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Color-Coding

Presented by Yuval Shimron Course 236801 1.12.2010. Color-Coding. The Uses of Color-Coding. Find solutions to sub-cases of the Subgraph Isomorphism Problem in polynomial time. Find More efficient solutions to some sub-cases that already had polynomial time solutions.

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Color-Coding

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  1. Presented by Yuval Shimron Course 2368011.12.2010 Color-Coding

  2. The Uses of Color-Coding • Find solutions to sub-cases of the Subgraph Isomorphism Problem in polynomial time. • Find More efficient solutions to some sub-cases that already had polynomial time solutions. • Find simple paths and cycles of specific length k. • This was the initial goal of the authors…

  3. Main Three New Results • (1) For a fixed k, if G=(V,E) contains a cycle of length k it can be found in O(Vω) expected time or O(VωlogV) worst-case time (ω<2.376 is the exponent of matrix multiplication). • (2) For a fixed k, if a planar graph G=(V,E) contains a cycle of length k it can be found in O(V) expected time or O(VlogV) worst-case time (Applies also to any non-trivial minor-closed family of graphs).

  4. Main Three New Results • (3) If G=(V,E) contains a subgraph isomorphic to a bounded tree-width graph H=(VH,EH) where |VH| = O(logV), then such a subgraph can be found in polynomial time. • Was not previously known even if H were just a simple path of length O(logV). • Shows that the LOG PATH problem is in NC (and not just in P).

  5. 1. The Method • Randomized method • Vertices are randomly colored using k = |VH| colors. • If |VH| = O(logV), then with a small (but only polynomial small) probability all the vertices of the (isomorphic to H) subgraph are colored in distinct colors. • Makes the task of finding this ‘color-coded’ subgraph much easier. • Be patient…

  6. 1. The Method • De-randomized algorithm? • Needs a family of colorings of G, such that every subset of k vertices of G is assigned with distinct colors by at least one of these coloring. • In other words, a family of perfect hash functions from {1, 2, …, |V|} to {1, 2, …, k}. • Only “small” loss of efficiency.

  7. 2. Random Orientations 2.1: Simple Path of Length k • If acyclic – simple • O(E) time for a simple algorithm. • So eliminate cycles: • Choose a random permutation . • Build by using : • Direct the edges:

  8. 2. Random Orientations 2.1: Simple Path of Length k • Every directed path of length k in G’ is a simple path of length k in G. • Every simple path of length k in G has a 2/(k+1)! chance of becoming a directed path in G’. • So if no path of length k was found in G’ repeat the process. • The expected number of times this process is repeated is at most (k+1)!/2.

  9. 2. Random Orientations 2.1: Simple Path of Length k • So we get O(E(k+1)!) time complexity. • This is also the result for the directed case. • Delete edges that don’t agree with . • Use the following fact + DFS to reduce it to O(V(k+1)!) for the undirected case: • Every graph with V vertices and at least k|V| edges contains a path of length k. • So first run a DFS on the original graph. • Apply the above algorithm only if no vertex of depth k was found (answered in O(k|V|) time).

  10. 2. Random Orientations 2.2: Simple Cycle of Length k • Choose random acyclic orientation G’. • Raise the adjacency matrix of G’ to the power of k-1 using O(logk) matrix multiplications. • This gives all the pairs of vertices connected by a path of length k-1. • Check if any of these pairs are connected. • If so . • If not, repeat the process. • Expected number of at most k!/2 time. • Complexity: O(k!(logk)Vω)=O(Vω) for a fixed k.

  11. 3. Color-Coding • To find a path of length k-1 in a graph G we can choose a random coloring of the vertices of G in k colors. • Every simple path of length k-1 in G has a chance of k!/kk > e-kto become colorful. • Each vertex is colored with a different color. • We can find it using lemma 3.1.

  12. 3. Color-Coding3.1: Colorful path of Length k-1 • Use Color-Coding to find a colorful path of length k-1 in 2O(k)E worst case time (if exists). • Actually it finds a path of length k that starts at a specific vertex s. • but we can always add some vertex s to G (with a new color). • The algorithm uses a given (random) coloring c :V {1, 2, … k} • The algorithm uses a dynamic programming approach.

  13. 3. Color-Coding3.1: Colorful path of Length k-1 • Suppose we’ve found for each vertex v the sets of colors on colorful paths of length i that connects s and v. • A collection of at most color sets. • For that we only need to record the color sets appearing on i-length paths. • And not the path themselves… • We inspect every color set C of that collection.

  14. 3. Color-Coding3.1: Colorful path of Length k-1 • We also inspect every edge (v,u) in E. • If we add to the collection of u that corresponds to colorful paths of length i+1. • The graph G contains a colorful path of length k-1iff the final collection, corresponds to paths of length k-1, of at least one vertex is non-empty.

  15. 3. Color-Coding 3.1: Colorful path of Length k-1 • The number of operations is at most . • The proof holds for both directed and undirected graphs.

  16. 3.2: Pairs of Vertices Connected by Colorful Paths of Length k-1 • We can find all pairs of vertices connected by path of length k-1 in or worst case time. • To get time simply run 3.1 algorithm |V| times, from each vertex of G=(V,E). • Use recursive approach to get time.

  17. 3.2: Pairs of Vertices Connected by Colorful Paths of Length k-1 • Keep all partitions of {1,2,…,k} into two subsets C1,C2of size k/2 each. There are such partitions. • For each partition, split G into two graphs derived from C1, C2 coloring. • Recursively find pairs of vertices connected by paths of k/2-1. • Store the results in Boolean matrices A1,A2.

  18. 3.2: Pairs of Vertices Connected by Colorful Paths of Length k-1 • Define B to be a Boolean matrix of adjacency relations between V1,V2 vertices. • Compute A1BA2. • You get all pairs connected by paths of length k-1 • First k/2 vertices are colored by colors from C1 • Last k/2 vertices are colored by colors from C2 • By OR-ingall the matrices obtained from all the partitions you get your answer. • Time complexity?

  19. 3. Color Coding3.3,3.4: Two Interesting Results • A simple path of length k-1 in a directed / undirected graph G=(V,E) can be found (if exists) in: • expected time for undirected graph. • DFS… • expected time for directed graph. • A simple cycle of size k in a directed / undirected graph G=(V,E) can be found (if exists) in either or expected time. • Simply use lemma 3.2.

  20. 4. Derandomized Orientations and Colorings • The previous randomized algorithms can be derandomized with a loss of efficiency. • Extra logV factor to the complexity. • What we need is a family of k-perfect hash functions from {1, 2, …, |V|} to {1, 2, …, k}. • If we use these hash functions we know that for every subset of k vertices there exists a coloring that gives each vertex in it, a distinct color.

  21. 4. Derandomized Orientations and Colorings • There exists an algorithm that constructs a k-perfect family of hash functions from{1, 2, ..., n} to {1, 2, ..., k}. • But its size is . • There also exists an algorithm that constructs a k-perfect family of hash functions from {1, 2, ..., n} to {1, 2, ..., k2} that its size is .

  22. 4. Derandomized Orientations and Colorings • So we use 2-level hashing: • Mapping from {1, 2, ..., n} to {1, 2, ..., k2} by using the second algorithm. • Mapping from {1, 2, ..., k2} to {1, 2, ..., k} by using the first algorithm.. • And we get just the promised extra O(logV) time. • The value of each element can be evaluated in O(1) time.

  23. 4. Derandomizing: Derandomize Orientations • Use k-perfect hash coloring functions. • Choose a random coloring (ant not a permutation) c :V --> {1, 2, … k} • Remove edges (u,v) s.t. . • Direct remaining edges (u,v) from u to v. • Again G’, the obtained graph, is acyclic. • Simple path of length k in G has a probability of 2k-k to become a directed path in G’. • Different from the Color-Coding method.

  24. 5. Cycles in Minor-ClosedFamilies of Graphs • An undirected graph G is d-degenerate if every subgraph of it has a vertex of degree at most d. • Smallest such d is called the degeneracy or the max-min degree of G. • Maximum over the minimum degrees of all sub-graphs of G. • If G is d-degenerate then clearly .

  25. 5. Cycles in Minor-Closed graphs5.1: Finding an Orientation • Let G be a connected undirected graph. • An acyclic orientation of G=(V,E) such that for every v we have can be found in O(E) time.

  26. 5. Minor-Closed Families of Graphs • A graph H is a minor of undirected graph G if it can be obtained from G by the removal and the contraction of edges. • A family C of graphs is minor-closed if a minor of any graph in it is also a member of the family. • If such C is non-trivial then all graphs in C are of bounded degeneracy. • s.t. .

  27. 5. Minor-Closed Families of Graphs • Consider the family of planar graphs Cplanar • It is minor-closed. • Each planar graph has a vertex whose degree is at most 5. • .

  28. 5. Minor-Closed Families of Graphs5.2: Simple Cycle of size k • Let C be a non-trivial minor-closed family of graphs and let be a fixed parameter.There exists a randomized algorithm that given an undirected graph in C finds a Ck - cycle of size k in it if one exists, in O(V) expected time. • Proof: • Let G = (V,E) be a graph in C that contains a Ck. • Choose a random coloring c : V -> {1, 2, 3, …, k}. • Ck is considered well-colored if colored in a consecutive way by the colors 1, 2, …, k.

  29. 5. Minor-Closed Families of Graphs5.2: Simple Cycle of size k • The Ck in G has a chance of 2/kk-1 to be well-colored. • Can we find it efficiently? • Yes, but with some probability… • Assume that the degeneracy of C is d = O(1). • We describe a randomized algorithm that given a coloring c, finds Ck with probability of 1/(2d)k. • Combining both gives a probability of at least so the expected time is .

  30. 5. Minor-Closed Families of Graphs5.2: Simple Cycle of size k • We can assume all edges of G connect vertices that are colored by consecutive colors (mod k). • Edges that don’t may be safely removed. • We orientate the graph so that the out-degree of all the vertices is at most d. • This takes only O(V) time. • The algorithm tries to find the edge that connects the vertices in Ck colored by k and k-1: vk ,vk-1. It “flipps coins” to guess it’s orientation and index – 2d possible combinations.

  31. 5. Minor-Closed Families of Graphs5.2: Simple Cycle of size k • For each guess of such index i • If the orientation is from vk-1 to vk: • All edges that leave vk-1 but whose index is not i are removed. • Otherwise does the opposite. • (for edges that leave vk) • Result is the graph G’ that contains a Ck with a probability of at least 1/(2d). • A forest of rooted stars.

  32. 5. Minor-Closed Families of Graphs5.2: Simple Cycle of size k • Each such star is contracted into a single vertex and assigned with the color k-1. • The obtained graph is denoted by G’’. • G’’ contains a well-colored Ck-1iffG’ contains a well-colored Ck. • Since each edge of G’ and therefore G’’ connects consecutively colored vertices. • G’’ is also a graph in the minor-closed family C. • So we recursively look for Ck-1.

  33. 5. Minor-Closed Families of Graphs5.2: Simple Cycle of size k • It will take us O((k-1)V) expected time. • And yields Ck-1 with a probability of at least 1/(2d)k-1. • Obviously it’s easy to reconstruct Ck from Ck-1. • We can stop the recursion when k=3 and use an existing algorithm for finding triangles in a general graph in time. • Any triangle in a three-colored graph is well-colored. • is in our case.

  34. 5.3: Simple Cycle of size kDerandomized Algorithm • There exists a determinist algorithm that given a graph in C, finds Ck if exist, in O(VlogV) WC time. • Proof: • Instead of using random coloring we exhaust a list of kO(k)logV colorings that has this property: • Every sequence of k vertices is consecutively colored by 1,2,…,k by at list one coloring of the list. • Instead of guessing the direction and index of each edge in the Ck we exhaust for each coloring all the (2d)kpossible choices. • If G contains a Ck then at least one Ck will be found this way.

  35. 6. Finding Bounded Tree-Width Sub-Graphs • A graph G1 is said to be isomorphic to a graph G2 if there exists a bijection: f : V(G1) -> V(G2) such that any two vertices u and v of G1 are adjacent in G1iff ƒ(u) and ƒ(v) are adjacent in G2.

  36. 6.1: Finding Sub-Graph Isomorphicto a Forest • Let F be a directed/undirected forest on k vertices. Let G be a directed/undirected graph. • A sub-graph isomorphic to F can be found if exists in: • expected time in the directed case. • expected time in the undirected case.

  37. 6.1: Finding Sub-Graph Isomorphicto a Forest • Proof: • Start as usual, by choosing a random coloring:c : V -> {1, …, k} of G. • With a probability of at least e-k the copy of F in G becomes colorful. • Meaning, each vertex is assigned with a different color. • Suppose that F is composed of l (directed) trees T1, T2, …, Tl with k1, k2, …, kl vertices each. • Let Fi be the (directed) forest composed of T1, T2, …, Ti.

  38. 6.1: Finding Sub-Graph Isomorphicto a Forest • For each we find the color sets that appear on colorful copies of Ti in G. • Note that copies of Ti , Tjwith disjoint color sets are necessarily disjoint. • Then, in 2O(k)time we find the color sets that appear on colorful copies of Fifor . • If the collection corresponding to F=Fl is not empty then G contains a colorful copy of F. • How do we find it…?

  39. 6.1: Finding Sub-Graph Isomorphicto a Forest • How dowe find the color sets that appear on colorful copies of Ti in G? • Let t be an arbitrary vertex in Ti=T. • For each vertex v in G we find the color sets that appear on copies of T in which v plays the role of r. • If T is a singe vertex then it’s easily done… • Otherwise let e=(r,r’) be a (directed) edge in T. • We break T into two (directed) sub-trees T’, T’’.

  40. 6.1: Finding Sub-Graph Isomorphicto a Forest • We recursively find, for each vertex v in G, the color sets in copies of T’ and T’’ in which v plays the role of r and then of r’. • For every (directed) edge (u,v) we update u’s collection with v’s collection if they are disjoint. • The complexity of this recursive algorithm is as required. • For the undirected case we use the fact that a graph with at least k|V| edges contains as a subgraph any forest on k vertices.

  41. 6.2: Tree Decomposition • Remember tree-width of a graph G? • The minimum tree-width over all possible tree-decomposition of G to (X,T). • T = (I, F) is a tree. • X = { Xi : iI} is a set of subsets of V such that: • The union of all Xis equals to V. • For every edge (u,v) of G there exists an i such that u,v are in Xi. • If , and j is on the path from i to k in T then:

  42. 6.3: Finding Sub-Graph Isomorphicto a Bounded TW Graph • Let H be a directed or undirected graph on k vertices with tree-width t. Let G be a directed or undirected graph. • A sub-graph of G, isomorphic to H, if one exists, can be found in expected time and in worst case time. • Proof is similar to that of Theorem 6.1. • So we will skip it...

  43. 6.3: Finding Sub-Graph Isomorphicto a Bounded TW Graph • In [RS86b] it is shown that if C is a minor closed family of graphs that excludes at least one planar graph G’ then there exists a (huge) constant cG’ such that every graph in C has a tree-width of at most cG’. • So we can use 6.3 wherever |VH| = O(logV) and H excludes at least one planar graph. • and decide in polynomial time whether G contains a graph isomorphic to H.

  44. Using 6.3: LOG PATH Problem • As a very special case of Theorem 6.3 we get that the LOG PATH problem is in P • A path of logV vertices is a tree. • In addition, all the algorithms we described are easily parallelizable. • So we get that the LOG PATH problem and other problems are in NC.

  45. Conclusions • The Color-Coding method efficiently finds k-vertex simple paths, k-vertex cycles, and other small sub-graphs within a given graph using probabilistic algorithms. • The Color-Coding method is a good example of demonstrating de-randomization techniques. • Algorithms presented can be easily parallelized. • Yielding efficient NC algorithms.

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