Re: [algogeeks] Re: Directi question-centre of the tree
farthest from 2: Find a vertex v1 | the farthest form r. 3: Find a vertex v2 | the farthest form v1. won't v2 be farthest from r? or we are talking about alternate pats also Best Regards Ashish Goel Think positive and find fuel in failure +919985813081 +919966006652 On Wed, Jun 20, 2012 at 6:25 PM, adarsh kumar algog...@gmail.com wrote: As you traverse along and find the diameter of the tree, keep track of the number of nodes thereby traversed. Let that be equal to n. Now, centre is the node corresponding to floor((n+1)/2). On Wed, Jun 20, 2012 at 5:19 PM, Nishant Pandey nishant.bits.me...@gmail.com wrote: I am asking again .. can any one please suggest better method for getting the median on the longest path of the tree .. efficient method . On Tue, Jun 19, 2012 at 5:08 PM, Nishant Pandey nishant.bits.me...@gmail.com wrote: for getting diameter we can simply add the max height of left subtree and max height of the right sub tree . the main issue is how efficiently we find median on that longest path to get the center of the tree . can any bdy sugest optimized algo for that ? On Mon, Jun 18, 2012 at 6:08 PM, rajesh pandey rajesh.pandey.i...@gmail.com wrote: I found it in some paper ;) Diameter and center De nition 4. The diameter of tree is the length of the longest path. De nition 5. A center is a vertex v such that the longest path from v to a leaf is minimal over all vertices in the tree.Tree center(s) can be found using simple algorithm. Algorithm 1. (Centers of tree) 1: Choose a random root r. 2: Find a vertex v1 | the farthest form r. 3: Find a vertex v2 | the farthest form v1. 4: Diameter is a length of path from v1 to v2. 5: Center is a median element(s) of path from v1 to v2. This is O(n) algorithm. It is clear that we can't determine tree isomorphism faster than O(n). So, if we nd a O(f(n)) algorithm for rooted trees isomorphism we can also obtain O(f(n)) algorithm for ordinary trees. On Saturday, June 16, 2012 12:04:32 PM UTC+5:30, achala sharma wrote: I think this algorithm is used for calculating poset in graph. On Sat, Jun 16, 2012 at 3:04 AM, Hemesh Singh hemesh.mn...@gmail.comwrote: + 1 for DK's solution. Is that a standard algorithm coz I feel like I have heard it somewhere ?? On Mon, Aug 8, 2011 at 1:37 AM, DK divyekap...@gmail.com wrote: @KK: DFS and BFS are O(N) and Floyd Warshall is O(N^3). Could you please state how you can use the traversals directly to get the center? (And prove your correctness too?) The solution given by Wladimir ( expanded upon by me) is O(N) and uses (somewhat) the inverse of a BFS as a traversal. -- DK http://twitter.com/divyekapoor http://gplus.to/divyekapoor http://www.divye.in -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To view this discussion on the web visit https://groups.google.com/d/**msg/algogeeks/-/HnMOZtOrkqwJhttps://groups.google.com/d/msg/algogeeks/-/HnMOZtOrkqwJ. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscribe@ **googlegroups.com algogeeks%2bunsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/** group/algogeeks?hl=enhttp://groups.google.com/group/algogeeks?hl=en . -- Hemesh singh -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscribe@* *googlegroups.com algogeeks%2bunsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/** group/algogeeks?hl=enhttp://groups.google.com/group/algogeeks?hl=en . -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To view this discussion on the web visit https://groups.google.com/d/msg/algogeeks/-/BWplK7bCatMJ. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en. -- Cheers, Nishant Pandey |Specialist Tools Development | npan...@google.comgvib...@google.com | +91-9911258345 -- Cheers, Nishant Pandey |Specialist Tools Development | npan...@google.comgvib...@google.com | +91-9911258345 -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en. -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to
Re: [algogeeks] Re: Directi question-centre of the tree
@ashish it wont be...first we r finding one end from any node dat is r n den frm dat end we r traversing to other deepest end... it is possible dat r b d intermediate node...n distance from r to v2 is smaller than from r to v1 -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en.
Re: [algogeeks] Re: Directi question-centre of the tree
I am asking again .. can any one please suggest better method for getting the median on the longest path of the tree .. efficient method . On Tue, Jun 19, 2012 at 5:08 PM, Nishant Pandey nishant.bits.me...@gmail.com wrote: for getting diameter we can simply add the max height of left subtree and max height of the right sub tree . the main issue is how efficiently we find median on that longest path to get the center of the tree . can any bdy sugest optimized algo for that ? On Mon, Jun 18, 2012 at 6:08 PM, rajesh pandey rajesh.pandey.i...@gmail.com wrote: I found it in some paper ;) Diameter and center De nition 4. The diameter of tree is the length of the longest path. De nition 5. A center is a vertex v such that the longest path from v to a leaf is minimal over all vertices in the tree.Tree center(s) can be found using simple algorithm. Algorithm 1. (Centers of tree) 1: Choose a random root r. 2: Find a vertex v1 | the farthest form r. 3: Find a vertex v2 | the farthest form v1. 4: Diameter is a length of path from v1 to v2. 5: Center is a median element(s) of path from v1 to v2. This is O(n) algorithm. It is clear that we can't determine tree isomorphism faster than O(n). So, if we nd a O(f(n)) algorithm for rooted trees isomorphism we can also obtain O(f(n)) algorithm for ordinary trees. On Saturday, June 16, 2012 12:04:32 PM UTC+5:30, achala sharma wrote: I think this algorithm is used for calculating poset in graph. On Sat, Jun 16, 2012 at 3:04 AM, Hemesh Singh hemesh.mn...@gmail.comwrote: + 1 for DK's solution. Is that a standard algorithm coz I feel like I have heard it somewhere ?? On Mon, Aug 8, 2011 at 1:37 AM, DK divyekap...@gmail.com wrote: @KK: DFS and BFS are O(N) and Floyd Warshall is O(N^3). Could you please state how you can use the traversals directly to get the center? (And prove your correctness too?) The solution given by Wladimir ( expanded upon by me) is O(N) and uses (somewhat) the inverse of a BFS as a traversal. -- DK http://twitter.com/divyekapoor http://gplus.to/divyekapoor http://www.divye.in -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To view this discussion on the web visit https://groups.google.com/d/* *msg/algogeeks/-/HnMOZtOrkqwJhttps://groups.google.com/d/msg/algogeeks/-/HnMOZtOrkqwJ. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscribe@** googlegroups.com algogeeks%2bunsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/** group/algogeeks?hl=en http://groups.google.com/group/algogeeks?hl=en . -- Hemesh singh -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscribe@** googlegroups.com algogeeks%2bunsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/** group/algogeeks?hl=en http://groups.google.com/group/algogeeks?hl=en. -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To view this discussion on the web visit https://groups.google.com/d/msg/algogeeks/-/BWplK7bCatMJ. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en. -- Cheers, Nishant Pandey |Specialist Tools Development | npan...@google.comgvib...@google.com | +91-9911258345 -- Cheers, Nishant Pandey |Specialist Tools Development | npan...@google.comgvib...@google.com | +91-9911258345 -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en.
Re: [algogeeks] Re: Directi question-centre of the tree
As you traverse along and find the diameter of the tree, keep track of the number of nodes thereby traversed. Let that be equal to n. Now, centre is the node corresponding to floor((n+1)/2). On Wed, Jun 20, 2012 at 5:19 PM, Nishant Pandey nishant.bits.me...@gmail.com wrote: I am asking again .. can any one please suggest better method for getting the median on the longest path of the tree .. efficient method . On Tue, Jun 19, 2012 at 5:08 PM, Nishant Pandey nishant.bits.me...@gmail.com wrote: for getting diameter we can simply add the max height of left subtree and max height of the right sub tree . the main issue is how efficiently we find median on that longest path to get the center of the tree . can any bdy sugest optimized algo for that ? On Mon, Jun 18, 2012 at 6:08 PM, rajesh pandey rajesh.pandey.i...@gmail.com wrote: I found it in some paper ;) Diameter and center De nition 4. The diameter of tree is the length of the longest path. De nition 5. A center is a vertex v such that the longest path from v to a leaf is minimal over all vertices in the tree.Tree center(s) can be found using simple algorithm. Algorithm 1. (Centers of tree) 1: Choose a random root r. 2: Find a vertex v1 | the farthest form r. 3: Find a vertex v2 | the farthest form v1. 4: Diameter is a length of path from v1 to v2. 5: Center is a median element(s) of path from v1 to v2. This is O(n) algorithm. It is clear that we can't determine tree isomorphism faster than O(n). So, if we nd a O(f(n)) algorithm for rooted trees isomorphism we can also obtain O(f(n)) algorithm for ordinary trees. On Saturday, June 16, 2012 12:04:32 PM UTC+5:30, achala sharma wrote: I think this algorithm is used for calculating poset in graph. On Sat, Jun 16, 2012 at 3:04 AM, Hemesh Singh hemesh.mn...@gmail.comwrote: + 1 for DK's solution. Is that a standard algorithm coz I feel like I have heard it somewhere ?? On Mon, Aug 8, 2011 at 1:37 AM, DK divyekap...@gmail.com wrote: @KK: DFS and BFS are O(N) and Floyd Warshall is O(N^3). Could you please state how you can use the traversals directly to get the center? (And prove your correctness too?) The solution given by Wladimir ( expanded upon by me) is O(N) and uses (somewhat) the inverse of a BFS as a traversal. -- DK http://twitter.com/divyekapoor http://gplus.to/divyekapoor http://www.divye.in -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To view this discussion on the web visit https://groups.google.com/d/ **msg/algogeeks/-/HnMOZtOrkqwJhttps://groups.google.com/d/msg/algogeeks/-/HnMOZtOrkqwJ. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscribe@* *googlegroups.com algogeeks%2bunsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/** group/algogeeks?hl=enhttp://groups.google.com/group/algogeeks?hl=en . -- Hemesh singh -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscribe@** googlegroups.com algogeeks%2bunsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/** group/algogeeks?hl=en http://groups.google.com/group/algogeeks?hl=en . -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To view this discussion on the web visit https://groups.google.com/d/msg/algogeeks/-/BWplK7bCatMJ. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en. -- Cheers, Nishant Pandey |Specialist Tools Development | npan...@google.comgvib...@google.com | +91-9911258345 -- Cheers, Nishant Pandey |Specialist Tools Development | npan...@google.comgvib...@google.com | +91-9911258345 -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en. -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en.
Re: [algogeeks] Re: Directi question-centre of the tree
for getting diameter we can simply add the max height of left subtree and max height of the right sub tree . the main issue is how efficiently we find median on that longest path to get the center of the tree . can any bdy sugest optimized algo for that ? On Mon, Jun 18, 2012 at 6:08 PM, rajesh pandey rajesh.pandey.i...@gmail.com wrote: I found it in some paper ;) Diameter and center De nition 4. The diameter of tree is the length of the longest path. De nition 5. A center is a vertex v such that the longest path from v to a leaf is minimal over all vertices in the tree.Tree center(s) can be found using simple algorithm. Algorithm 1. (Centers of tree) 1: Choose a random root r. 2: Find a vertex v1 | the farthest form r. 3: Find a vertex v2 | the farthest form v1. 4: Diameter is a length of path from v1 to v2. 5: Center is a median element(s) of path from v1 to v2. This is O(n) algorithm. It is clear that we can't determine tree isomorphism faster than O(n). So, if we nd a O(f(n)) algorithm for rooted trees isomorphism we can also obtain O(f(n)) algorithm for ordinary trees. On Saturday, June 16, 2012 12:04:32 PM UTC+5:30, achala sharma wrote: I think this algorithm is used for calculating poset in graph. On Sat, Jun 16, 2012 at 3:04 AM, Hemesh Singh hemesh.mn...@gmail.comwrote: + 1 for DK's solution. Is that a standard algorithm coz I feel like I have heard it somewhere ?? On Mon, Aug 8, 2011 at 1:37 AM, DK divyekap...@gmail.com wrote: @KK: DFS and BFS are O(N) and Floyd Warshall is O(N^3). Could you please state how you can use the traversals directly to get the center? (And prove your correctness too?) The solution given by Wladimir ( expanded upon by me) is O(N) and uses (somewhat) the inverse of a BFS as a traversal. -- DK http://twitter.com/divyekapoor http://gplus.to/divyekapoor http://www.divye.in -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To view this discussion on the web visit https://groups.google.com/d/** msg/algogeeks/-/HnMOZtOrkqwJhttps://groups.google.com/d/msg/algogeeks/-/HnMOZtOrkqwJ. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscribe@** googlegroups.com algogeeks%2bunsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/** group/algogeeks?hl=en http://groups.google.com/group/algogeeks?hl=en. -- Hemesh singh -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscribe@** googlegroups.com algogeeks%2bunsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/** group/algogeeks?hl=en http://groups.google.com/group/algogeeks?hl=en. -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To view this discussion on the web visit https://groups.google.com/d/msg/algogeeks/-/BWplK7bCatMJ. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en. -- Cheers, Nishant Pandey |Specialist Tools Development | npan...@google.comgvib...@google.com | +91-9911258345 -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en.
Re: [algogeeks] Re: Directi question-centre of the tree
I found it in some paper ;) Diameter and center De nition 4. The diameter of tree is the length of the longest path. De nition 5. A center is a vertex v such that the longest path from v to a leaf is minimal over all vertices in the tree.Tree center(s) can be found using simple algorithm. Algorithm 1. (Centers of tree) 1: Choose a random root r. 2: Find a vertex v1 | the farthest form r. 3: Find a vertex v2 | the farthest form v1. 4: Diameter is a length of path from v1 to v2. 5: Center is a median element(s) of path from v1 to v2. This is O(n) algorithm. It is clear that we can't determine tree isomorphism faster than O(n). So, if we nd a O(f(n)) algorithm for rooted trees isomorphism we can also obtain O(f(n)) algorithm for ordinary trees. On Saturday, June 16, 2012 12:04:32 PM UTC+5:30, achala sharma wrote: I think this algorithm is used for calculating poset in graph. On Sat, Jun 16, 2012 at 3:04 AM, Hemesh Singh hemesh.mn...@gmail.comwrote: + 1 for DK's solution. Is that a standard algorithm coz I feel like I have heard it somewhere ?? On Mon, Aug 8, 2011 at 1:37 AM, DK divyekap...@gmail.com wrote: @KK: DFS and BFS are O(N) and Floyd Warshall is O(N^3). Could you please state how you can use the traversals directly to get the center? (And prove your correctness too?) The solution given by Wladimir ( expanded upon by me) is O(N) and uses (somewhat) the inverse of a BFS as a traversal. -- DK http://twitter.com/divyekapoor http://gplus.to/divyekapoor http://www.divye.in -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To view this discussion on the web visit https://groups.google.com/d/msg/algogeeks/-/HnMOZtOrkqwJ. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en. -- Hemesh singh -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en. -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To view this discussion on the web visit https://groups.google.com/d/msg/algogeeks/-/BWplK7bCatMJ. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en.
Re: [algogeeks] Re: Directi question-centre of the tree
I think this algorithm is used for calculating poset in graph. On Sat, Jun 16, 2012 at 3:04 AM, Hemesh Singh hemesh.mn...@gmail.comwrote: + 1 for DK's solution. Is that a standard algorithm coz I feel like I have heard it somewhere ?? On Mon, Aug 8, 2011 at 1:37 AM, DK divyekap...@gmail.com wrote: @KK: DFS and BFS are O(N) and Floyd Warshall is O(N^3). Could you please state how you can use the traversals directly to get the center? (And prove your correctness too?) The solution given by Wladimir ( expanded upon by me) is O(N) and uses (somewhat) the inverse of a BFS as a traversal. -- DK http://twitter.com/divyekapoor http://gplus.to/divyekapoor http://www.divye.in -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To view this discussion on the web visit https://groups.google.com/d/msg/algogeeks/-/HnMOZtOrkqwJ. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en. -- Hemesh singh -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en. -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en.
Re: [algogeeks] Re: Directi question-centre of the tree
+ 1 for DK's solution. Is that a standard algorithm coz I feel like I have heard it somewhere ?? On Mon, Aug 8, 2011 at 1:37 AM, DK divyekap...@gmail.com wrote: @KK: DFS and BFS are O(N) and Floyd Warshall is O(N^3). Could you please state how you can use the traversals directly to get the center? (And prove your correctness too?) The solution given by Wladimir ( expanded upon by me) is O(N) and uses (somewhat) the inverse of a BFS as a traversal. -- DK http://twitter.com/divyekapoor http://gplus.to/divyekapoor http://www.divye.in -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To view this discussion on the web visit https://groups.google.com/d/msg/algogeeks/-/HnMOZtOrkqwJ. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en. -- Hemesh singh -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en.
Re: [algogeeks] Re: Directi question-centre of the tree
Traverse the tree inorder. Store the values in an array. Find the element in the middle of the array. On Sun, Aug 7, 2011 at 8:57 AM, subramania jeeva subramaniaje...@gmail.comwrote: 5 /\ 6 7 / 8 Here the centre is both 5 and 6 . we need to find both of them..:) Cheers ~ Jeeva ~ -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en. -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en.
Re: [algogeeks] Re: Directi question-centre of the tree
@Everyone: Wladimir has posted the correct solution to the problem and it is
an O(N) bottom up solution.
*The original solution:*
On Wednesday, 6 October 2010 21:10:40 UTC+5:30, wladimirufc wrote:
Find the leaf of tree then put all leaf in a queue.
While queue not empty:
remove u from queue
remove u of tree
if some v adjacent a u become leaf
insert v in queue
the last vertice is the center of the tree
Let me expand on the solution for a bit.
*Definition:*
1. For the purposes of this solution, a *leaf node* is defined as a node
having only one edge connected to it.
*Visualization:*
Think about the tree being flattened onto a table and laid out in a circular
fashion.
eg.
1
/ / \ \
4 2 3 7
/\
5 6
\
8
can be laid out as:
4 3
\ /
1 -- 7
|
2
/\
5 6
\
8
Now, wrap a rubber band around the layout and start tightening it.
At every stage, you'll be getting rid of nodes with only a single edge (so
called leaf nodes)
Stop when you end up with either a single node or a pair of nodes connected
by an edge.
*For example, *
remove leaf nodes as:
Step 1: 8,5,4,3,7
Step 2: 6,1 (Note: 2 is not removed as it has 2 connecting edges)
The center of the tree is thus 2.
*Proof of correctness:*
Let T be a tree for which D(u, v) = D(v, u) = C is the maximum possible.
It is obvious that both u and v must be leaf nodes since u..v is a diameter
of the graph.
Also, any non-leaf node w on the path u...v has M(w) M(u) or M(v) since
D(u, v) = D(u, w) + D(w, v)
Consider all such diameters of the tree graph T.
Let the pairs (u1, v1), (u2, v2)... (uk, vk) represent the nodes of a
diameter of the graph.
Consider the subgraph G' = G - {u1, u2, ..., uk, v1, v2, .. vk}
The diameter of G' is C-2. The center(s) of the graph is/are unchanged.*
Applying this reduction multiple times, any tree's undirected graph
representation
can be reduced to a graph of diameter 0 or 1.
*Base cases: *
Graph of diameter 0: The node itself is the center.
Graph of diameter 1: The pair of connected nodes are the center of the
graph.
*Note:* The connectedness of the graph is maintained at every stage of the
reduction process as no bridge nodes are ever removed.
** Proof of reduction of diameter by 2:*
Let us assume that the diameter of G' is C-1 with the nodes making up the
diameter being (p, q).
Therefore atleast one of p and q is a leaf node in G.
But since we removed all the leaf nodes in the reduction process, both p and
q must be internal nodes.
Hence the diameter of G' cannot be C-1.
By removing the 2 end nodes on a diameter of G, we are assured that there
exists a path of length C-2 in G'.
Since the diameter is an integral value and G' cannot have a diameter =
C-1, therefore the diameter
of G' is C-2.
** Proof of non-changing nature of the center:* *(slightly hand-wavy, I
would appreciate slightly more formality)*
Since the reduced graph G' has all the nodes of the original diameter
(except the leaf nodes) present and the
diameter of the graph G is also present as one of the diameters of graph G',
the center of the graph is
still present in G'.
Taking all these parts together proves the result and also provides the
algorithm for obtaining the result
in O(N).
Hope you enjoyed it!
--
DK
http://gplus.to/divyekapoor
http://twitter.com/divyekapoor
http://www.divye.in
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[algogeeks] Re: Directi question-centre of the tree
Codechef Ques(Initiative of Directi) use DFS, BFS or Floyd Warshall... :) -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en.
[algogeeks] Re: Directi question-centre of the tree
@KK: DFS and BFS are O(N) and Floyd Warshall is O(N^3). Could you please state how you can use the traversals directly to get the center? (And prove your correctness too?) The solution given by Wladimir ( expanded upon by me) is O(N) and uses (somewhat) the inverse of a BFS as a traversal. -- DK http://twitter.com/divyekapoor http://gplus.to/divyekapoor http://www.divye.in -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To view this discussion on the web visit https://groups.google.com/d/msg/algogeeks/-/HnMOZtOrkqwJ. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en.
[algogeeks] Re: Directi question-centre of the tree
From any node, find the farthest node by DFS, save its location(say A). Now start DFS from A, and reach the farthest node, say B. AB will be diameter of tree and longest path. The central node will be the centre of tree. O(n) solution. On Jul 27, 1:53 am, sivaviknesh s sivavikne...@gmail.com wrote: we can do like creating an adjacency matrix..populate all values i.e. cost to reach each node...finally find the nodes that has min value...any other gud solutions?? On Wed, Jul 27, 2011 at 2:16 AM, sivaviknesh s sivavikne...@gmail.comwrote: the question is clear..you draw the sample test case and work out .. you will understand 1 / / \ 4 2 3 /\ 5 6 here from node 2 the maximum cost to reach any node 'i' is 2 and minimum is 1 ...so M(2) = max(1,2)=2..the same is the case for node 1 also ...if you consider node 4 , to reach 6 , cost is 3similar is applicable for other nodes therefore centre = min(M(1..6)) = 1 and 2. ...hope u got me!! On Wed, Jul 27, 2011 at 2:08 AM, siva viknesh sivavikne...@gmail.comwrote: -- Forwarded message -- From: jayapriya surendran priya7...@gmail.com Date: Sep 29 2010, 9:41 pm Subject: Directi question-centre of the tree To: Algorithm Geeks In graph theory, atreeis defined as a graph on N nodes,and (N-1) un-directed edges such that there are no cycles in the graph.Each node has a single unique path to every other node. Let D(u,v) be the number of edges in the unique path from node 'u' to node 'v' (or from node 'v' to 'u' since the edges are un-directed).D(u,u) is 0 for all nodes 'u'. M(u)=MAX(D(u,i):for all nodes i) Thecenterof atreeis the node (or nodes) 'u',for which M(u) is minimum among all the nodes in the graph. You'll be given a graph which has N nodes (1=N=20).The nodes are labeled 1,2,3,..N.You will be provided with N-1 edges in the form of a b pairs where 1=a,b=N.No edge will be repeated.You can assume that the edges are specified such that the graph is a validtreeas defined above. Output the node labels of thecenter(or centers) of thetree. Sample Input: 6(value of N) 1 3 (edges) 1 4 1 2 2 5 2 6 Sample Output 1 2 Expected:O(N) complexity algo can anyone plz help me out with O(N) algo? -- Regards, $iva -- Regards, $iva -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en.
Re: [algogeeks] Re: Directi question-centre of the tree
5 /\ 6 7 / 8 Here the centre is both 5 and 6 . we need to find both of them..:) Cheers ~ Jeeva ~ -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en.
[algogeeks] Re: Directi question-centre of the tree
the question is clear..you draw the sample test case and work out .. you will understand 1 / / \ 4 2 3 /\ 5 6 here from node 2 the maximum cost to reach any node 'i' is 2 and minimum is 1 ...so M(2) = max(1,2)=2..the same is the case for node 1 also ...if you consider node 4 , to reach 6 , cost is 3similar is applicable for other nodes therefore centre = min(M(1..6)) = 1 and 2. ...hope u got me!! On Wed, Jul 27, 2011 at 2:08 AM, siva viknesh sivavikne...@gmail.comwrote: -- Forwarded message -- From: jayapriya surendran priya7...@gmail.com Date: Sep 29 2010, 9:41 pm Subject: Directi question-centre of the tree To: Algorithm Geeks In graph theory, atreeis defined as a graph on N nodes,and (N-1) un-directed edges such that there are no cycles in the graph.Each node has a single unique path to every other node. Let D(u,v) be the number of edges in the unique path from node 'u' to node 'v' (or from node 'v' to 'u' since the edges are un-directed).D(u,u) is 0 for all nodes 'u'. M(u)=MAX(D(u,i):for all nodes i) Thecenterof atreeis the node (or nodes) 'u',for which M(u) is minimum among all the nodes in the graph. You'll be given a graph which has N nodes (1=N=20).The nodes are labeled 1,2,3,..N.You will be provided with N-1 edges in the form of a b pairs where 1=a,b=N.No edge will be repeated.You can assume that the edges are specified such that the graph is a validtreeas defined above. Output the node labels of thecenter(or centers) of thetree. Sample Input: 6(value of N) 1 3 (edges) 1 4 1 2 2 5 2 6 Sample Output 1 2 Expected:O(N) complexity algo can anyone plz help me out with O(N) algo? -- Regards, $iva -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en.
[algogeeks] Re: Directi question-centre of the tree
we can do like creating an adjacency matrix..populate all values i.e. cost to reach each node...finally find the nodes that has min value...any other gud solutions?? On Wed, Jul 27, 2011 at 2:16 AM, sivaviknesh s sivavikne...@gmail.comwrote: the question is clear..you draw the sample test case and work out .. you will understand 1 / / \ 4 2 3 /\ 5 6 here from node 2 the maximum cost to reach any node 'i' is 2 and minimum is 1 ...so M(2) = max(1,2)=2..the same is the case for node 1 also ...if you consider node 4 , to reach 6 , cost is 3similar is applicable for other nodes therefore centre = min(M(1..6)) = 1 and 2. ...hope u got me!! On Wed, Jul 27, 2011 at 2:08 AM, siva viknesh sivavikne...@gmail.comwrote: -- Forwarded message -- From: jayapriya surendran priya7...@gmail.com Date: Sep 29 2010, 9:41 pm Subject: Directi question-centre of the tree To: Algorithm Geeks In graph theory, atreeis defined as a graph on N nodes,and (N-1) un-directed edges such that there are no cycles in the graph.Each node has a single unique path to every other node. Let D(u,v) be the number of edges in the unique path from node 'u' to node 'v' (or from node 'v' to 'u' since the edges are un-directed).D(u,u) is 0 for all nodes 'u'. M(u)=MAX(D(u,i):for all nodes i) Thecenterof atreeis the node (or nodes) 'u',for which M(u) is minimum among all the nodes in the graph. You'll be given a graph which has N nodes (1=N=20).The nodes are labeled 1,2,3,..N.You will be provided with N-1 edges in the form of a b pairs where 1=a,b=N.No edge will be repeated.You can assume that the edges are specified such that the graph is a validtreeas defined above. Output the node labels of thecenter(or centers) of thetree. Sample Input: 6(value of N) 1 3 (edges) 1 4 1 2 2 5 2 6 Sample Output 1 2 Expected:O(N) complexity algo can anyone plz help me out with O(N) algo? -- Regards, $iva -- Regards, $iva -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algogeeks@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en.
[algogeeks] Re: Directi question-centre of the tree
I am thinking on these lines. Start from any node. DFS to the fartheset node from there. let the farthest node be B. Start a new DFS from B to reach the fartheset node from B , let that be C. It can be proved that BC is the longest path in the tree. Then, the node in the center will be the answer to the question. Incase the path length is even we will have two nodes. I havent proved it yet, the second part of my solution. this was a quick thought. On Sep 29, 9:41 pm, jayapriya surendran priya7...@gmail.com wrote: In graph theory, a tree is defined as a graph on N nodes,and (N-1) un-directed edges such that there are no cycles in the graph.Each node has a single unique path to every other node. Let D(u,v) be the number of edges in the unique path from node 'u' to node 'v' (or from node 'v' to 'u' since the edges are un-directed).D(u,u) is 0 for all nodes 'u'. M(u)=MAX(D(u,i):for all nodes i) The center of a tree is the node (or nodes) 'u',for which M(u) is minimum among all the nodes in the graph. You'll be given a graph which has N nodes (1=N=20).The nodes are labeled 1,2,3,..N.You will be provided with N-1 edges in the form of a b pairs where 1=a,b=N.No edge will be repeated.You can assume that the edges are specified such that the graph is a valid tree as defined above. Output the node labels of the center(or centers) of the tree. Sample Input: 6(value of N) 1 3 (edges) 1 4 1 2 2 5 2 6 Sample Output 1 2 Expected:O(N) complexity algo can anyone plz help me out with O(N) algo? -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algoge...@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en.
[algogeeks] Re: Directi question-centre of the tree
I don't understand what you mean when you say minimum among all the nodes in the graph. In any case, your definition of centre of tree looks similar to the closeness centrality measure - http://www.faculty.ucr.edu/~hanneman/nettext/C10_Centrality.html#Closeness I doubt you can do it in O(N)... On Sep 29, 12:41 pm, jayapriya surendran priya7...@gmail.com wrote: In graph theory, a tree is defined as a graph on N nodes,and (N-1) un-directed edges such that there are no cycles in the graph.Each node has a single unique path to every other node. Let D(u,v) be the number of edges in the unique path from node 'u' to node 'v' (or from node 'v' to 'u' since the edges are un-directed).D(u,u) is 0 for all nodes 'u'. M(u)=MAX(D(u,i):for all nodes i) The center of a tree is the node (or nodes) 'u',for which M(u) is minimum among all the nodes in the graph. You'll be given a graph which has N nodes (1=N=20).The nodes are labeled 1,2,3,..N.You will be provided with N-1 edges in the form of a b pairs where 1=a,b=N.No edge will be repeated.You can assume that the edges are specified such that the graph is a valid tree as defined above. Output the node labels of the center(or centers) of the tree. Sample Input: 6(value of N) 1 3 (edges) 1 4 1 2 2 5 2 6 Sample Output 1 2 Expected:O(N) complexity algo can anyone plz help me out with O(N) algo? -- You received this message because you are subscribed to the Google Groups Algorithm Geeks group. To post to this group, send email to algoge...@googlegroups.com. To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/algogeeks?hl=en.
