Noga Alon Institute for Advanced Study and Tel Aviv University Color-coding Shirly Zilkha Noga Alon Institute for Advanced Study and Tel Aviv University Raphael Yuster Dept. of Computer Science Tel Aviv University Uri Zwick Dept. of Computer Science Tel Aviv University

Talk Plan Backgroud, motivation Main results of the paper Previous results Definitions, constructions, proofs Techniques, ideas Open problems, future research

Backgroud, motivation Novel method for finding simple paths and cycles of a specified length k, and other small subgraphs, within a given graph G = (V,E). randomized method of color-coding can be derandomized using families of perfect hash functions.

Main results of the paper a simple cycle of size exactly k (for a fixed k) can be found in either O(Vω) expected time or O(Vωlog V ) worst-case time, where ω < 2.376 is the exponent of matrix multiplication. In a planar graph such a cycle can be found in either O(V ) expected time or O(V log V ) worst-case time. If a graph G = (V,E) contains a subgraph isomorphic to a bounded tree-width graph H = (VH,EH) where |VH| = O(log V ), then such a copy of H can be found in polynomial time. This was not previously known even if H were just a path of length O(log V ).

Previous results These results improve upon previous results of many authors. The third result resolves in the affirmative a conjecture of Papadimitriou and Yannakakis that the LOG PATH problem is in P. The paper shows that it is even in NC.

Simple paths of length k-1 Choose a random coloring of the vertices of G with k colors. Definition: path is colorful iff each vertex on it is colored by a distinct color. Colorful path is clearly simple Simple path has a chance of k!/kk > e-k to become colorful.

Colorful Path Lemma Lemma 3.1: For a given coloring, a colorful path of length k-1, if exists, can be found in 2O(k)·E worst-case time.

Colorful Path Lemma (Proof) The idea of the proof: first prove it for path starting from a specific vertex (using dynamic programming) Add a new vertex with a new color that is connected to all the vertices of G

Colorful Path Lemma (Proof 2) We keep in each vertex the colors of colorful paths connecting it to the start vertex 1 3 2 (3) ø

Colorful Path Lemma (Proof 2) Each vertex holds 2k entries 1 3 2 (3) (1,3) (2,3) ø

Colorful Path Lemma (Proof 2) We update recursively the neighbors of each vertex only if its color was not used by the path 1 3 2 (3) (1,3) (2,3) ø (2,3) (1,2,3)

Colorful Path Lemma (Proof 2) The graph G contains a colorful path of length k-1 with respect to the coloring c if and only if the final collection, that corresponding to paths of length k-1, of at least one vertex is non-empty. The number of operations performed by the algorithm outlined is at most O(Σii·(ik)·|E|) which is clearly O(k2k·E).

Finding connected vertices Lemma 3.2: For a given coloring, all pairs of vertices connected by colorful paths of length k-1, can be found in 2O(k)·V·E worst-case time or 2O(k)·Vω worst-case time

Finding connected vertices (Proof) Proof (2O(k)·V·E worst-case time ): Run the algorithm described in the proof of Lemma 3.1 |V| times, once for each starting vertex.

Finding connected vertices (Proof) Proof (2O(k)·Vω worst-case time ): Idea: If there is a colorful path, then there exist partition of colors such that the first part of the path is colored by colors of the first subgroup of colors and the second by second. So we will try all possible partitions of colors and look for a colorful path of half length by recursion. 2 1 3 4

Finding connected vertices (Proof) Proof (2O(k)·Vω worst-case time ): Enumerate all partitions of the color set {1,2… k} into two subsets C1,C2 of size k/2 ((kk/2) < 2k partitions) Ex:{({1,2}{3,4}), ({1,3}{2,4}), ({(1,4),(2,3)})} 2 1 3 4

Finding connected vertices (Proof) Proof (2O(k)·Vω worst-case time ): For each such partition C1,C2, let V1 be the set of vertices of G colored by colors from C1 and V2 be the set of vertices of G colored by colors from C2. Let G1 and G2 be the subgraphs of G induced by V1 and V2 respectively. Ex:C1={1,2} , c2={3,4} 2 1 3 4

Finding connected vertices (Proof) Proof (2O(k)·Vω worst-case time ): Recursively find all pairs of vertices in G1 and in G2 connected by colorful paths of length k/2-1.

Finding connected vertices (Proof) Proof (2O(k)·Vω worst-case time ): Collect this information into two Boolean matrices A1 and A2. Let B be a Boolean matrix that describes the adjacency relations between the vertices of V1 and those of V2. The Boolean product A1BA2 gives all pairs of vertices in V that are connected by colorful paths of length exactly k-1, where the first k/2 vertices on the paths are colored by colors from C1 and the last k/2 vertices are colored by colors from C2.

Finding connected vertices (Proof) Proof (2O(k)·Vω worst-case time ): By OR-ing the matrices obtained for all the partitions we obtain the desired result.

Finding connected vertices (Proof) Proof (2O(k)·Vω worst-case time ): It is easy to verify that the complexity of this approach is indeed 2O(k)·Vω ,as the number of matrix multiplication used, t(k), satisfies the recurrence t(k) < 2k t(k/2).

Theorem 3.3 A simple directed or undirected path of length k-1 in a (directed or undirected) graph G = (V,E) that contains such a path can be found in 2O(k)·E expected time.

Theorem 3.4 A simple directed or undirected cycle of size k in a (directed or undirected) graph G = (V,E) that contains such a cycle can be found in either 2O(k)·V·E or 2O(k)·Vω expected time.

Proof of the theorems We pick a random coloring of k colors. The simple path (cycle) has a chance of p=k!/kk to be colored in different colors The expected time until choosing a good coloring is 1/p<ek<22k 22k·2O(k) =2O(k)

Derandomized colorings The randomized algorithms of the previous two sections can be derandomized with only a small loss of eficiency. The 2O(k) dependence on k is retained for the small price of an extra logV factor to the complexity.

Derandomized colorings We want to give every simple path of length k-1 a chance of being discovered. We need a list of colorings of V such that for every subset of size k of V there exists a coloring in the list that gives each its vertex a distinct color. This is a k-perfect family of hash functions from {1,2,…,|V|} to {1,2,…,k}.

k-perfect hash family Definition: a k-perfect hash family of size s consists of a sequence φ1, φ2, …, φs of functions from {1,…,n} to {1,…,m} with the property, that for any k-subset X of {1,…,n}, there exists i such that φi is injective when restricted to X.

Derandomized colorings There is an explicit construction of a k-perfect family from {1,2,…,n} to {1,2,…,k} in which each function is specified using O(k) +2loglogn bits. The size of the family is therefore 2(O(k)+2loglogn) =2O(k)log2n. This gives us derandomization with factor of log2V (and not of logV as promised).

logV Derandomizing The size of the family of hash functions can be reduced to 2O(k)logn in the following way: construct a k-perfect family that maps {1,2,…,n} to {1,2,…,k2}. construct a k-perfect family that maps {1,2,…, k2} to {1,2,…,k}. compose these families of hash functions.

Derandomized colorings A k-perfect family of size 2O(k) from {1,2,…, k2} to {1,2,…,k} can be obtained using the previously mentioned construction. (2O(k)+log2k2=2O(k))

Derandomized colorings Now we want to construct a k-perfect family of size kO(1)logn from {1,2,…,n} to {1,2,…,k2} We use small probability spaces that support sequences of almost l-wise independent random variables.

Derandomized colorings Definition: A sequence X1,…,Xn of random Boolean variables is (ε,l)-independent if for any l positions i1 < i2 < ...< il and any l bits α1,…, αl we have |Pr[Xi1= α1,…,Xil= αl]-2-l| < ε If the X1,…,Xn are (2-l,l)-independent, then any subset of l variables attains each one of its 2l possible values with some positive probability.

Derandomized colorings There are constructions where the size of sample spaces that support n random variables that are (ε,l)-independent can be as small as 2O(l+log(1/ ε)logn and they can be constructed in 2O(l+log(1/ ε)nlogn time.

Derandomized colorings To construct a k-perfect family of size kO(1)logn from {1,2,…,n} to {1,2,…,k2} we use a probability space of size kO(1) logn that supports l·n random variables that are (2-2l,2·l)-independent, where l= 2logk (i.e. 2l=k2). We attach l random variables to each element of {1,2,…,n} thereby assigning it a color from {1,2,…,k2}.

Derandomized colorings The probability that two elements i and j are assigned the same color is at most 21-l =2/k2: Pr[Xi1= C1, Xi2= C2]-2-2l| < 2-2l Pr[Xi1= Xi2= C]< 2·2-2l Pr[Xi1= Xi2] = ΣCPr[Xi1= Xi2= C] < 2l·2·2-2l = 21-l

Derandomized colorings The probability that two distinct elements in a set of size k are assigned the same color is less than (k2)·(2/k2) < 1 (because it is less than the sum of probabilities of all pairs) So the probability of all k elements to be different is strictly positive. So the family is indeed k-perfect.

Open problems, future research Is there a polynomial time (deterministic or randomized) algorithm for deciding if a given graph contains a path of length log2V ? (The answer “NO” was proven later by Alon&Gutner) Can the logV factor appearing in our derandomization be omitted? Is the problem of deciding whether a given graph contains a triangle as dificult as the Boolean multiplication of two V by V matrices?

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