Balanced Hashing, Color Coding and Approximate Counting

Iterative Expansion and Color Coding

“…We remark that during the stage of the final revision of the current article, Alon and Gutner [2009] announced a construction of k-color coding scheme of nearly optimal size O(e k+O(log 3 k) log n).…”

Waring Rank, Parameterized and Exact Algorithms

“…, α π(n) ). This is equivalent to the color-coding algorithm for counting cycles described in [AG09], except we use inclusion-exclusion instead of dynamic programming to count the number of colorful simple cycles for a given coloring. Similarly, by replacing F with an (n, d)perfect hash family one obtains an algorithm for detecting simple cycles that parallels the one given in [AYZ95].…”

Maximum Disjoint Paths on Edge-Colored Graphs: Approximability and Tractability

“…Crucial to the overall running time is the size of a k-perfect family and the time required to enumerate and evaluate the hash functions of the family. Currently, the best bounds (such as [8,16,17]) are, in general, explicit constructions of families of size 2 O(k) log O(1) (|U |) in time proportional to their size.…”

“…, k} such that for each U ⊆ U of cardinality k there exists a hash function f ∈ F which assigns distinct integers to the elements of U . It has been shown (see, for example, [8,16,17]) that a k-perfect family of hash functions of size 2 O(k) log O(1) |U | can be explicitly constructed in time proportional to its size. As a consequence, the -LCDP p problem can be solved by solving the L-labelled -LCDP p problem for all the labelling functions given by the hash functions of a ( − 1)p-perfect family (where…”