A graph is a pair (V, E), where ... What is the Big-O of the following code? A.O (n) B.O (log_2 n) C.O (n ...

Basic algorithm analysis (e.g. big-O analysis of a graph algorithms). Basic probability theory (e.g. Bayes Rule). A genuine interest in language; linguistics background is a huge plus. Related courses. See also previous offerings of this course.

on deriving asymptotic (so called, \big-O") bounds on the space, query time, and cost of operations for a given data structure. On the practical side, you will be writing programs to implement a number of classical data structures. This will acquaint you with the skills needed to develop clean designs and debug them.

function of n and k. For partial credit give it in big-O notation.) (d)Consider the code a = h00i, b = h01i, c = h11i, d = h101i. (i)Is this a pre x code? Explain. (ii)Could this code be generated by Hu man’s algorithm? Explain. (e) G is a directed graph with negative weight edges but no negative weight cycles. Which

2. The graph is regular of degree qt 11 q 1 (which is asymptotically n 1 t). 3. NG q;tdoes not contain K t;t!+1. 4. The largest eigenvalue of the graph is qt 1 q 1, the absolute value of each other eigenvalue is bounded by (q 2)qt=2 q 1 (which is < p n). 5. The independence number (of the graph obtained from NG q;tby omitting all loops) is O(n ...

tices in a directed graph. The Floyd-Warshall algorithm dates back to the early 60’s. Warshall was interested in the weaker question of reachability: determine for each pair of vertices u and v, whether u can reach v. Floyd realized that the same technique could be used to compute shortest paths with only minor variations.

in this graph, that is, (x;y) 2R if and only if there exists a path from xto yin R. The Floyd-Warshall algorithm can be modi ed to compute the transitive closure in time O(n3), where n= jXj. All-Pairs Max-Capacity Paths: Let G= (V;E) be a directed graph with positive edge capacities c(u;v).

Clique (CLIQUE): The clique problem is: given an undirected graph G = (V;E) and an integer k, does G have a subset V0 of k vertices such that for each distinct u;v 2V0, fu;vg2E. In other words, does G have a k vertex subset whose induced subgraph is complete. Vertex Cover (VC): A vertex cover in an undirected graph G = (V;E) is a subset of

reasons the RAM model should really only be used for asymptotic (i.e. big-O) analysis. The RAM has also served as a tool to better understand algorithmic techniques such as divide-and-conquer, and dynamic programming, among many others. Finally, and importantly, it is natural to write pseudocode or real code that is naturally translated to the RAM.

The big Ramsey degree of a in S is the least integer t wuch that some subset S′order-equivalent to S where the coloring restricted to S′only uses t colors is guaranteed. Maˇsulovi´c and ˇSobot (2019) showed that all countable ordinals less than ωω have finite big Ramsey degrees. We find exact big Ramsey degrees for all ordinals less ...