I ACCEPT NO RESPONSIBILITY FOR ERRORS ON THIS SHEET. I assume that E = (V ).

Transcription

1 1 I ACCEPT NO RESPONSIBILITY FOR ERRORS ON THIS SHEET. I assume that E = (V ). Data structures Sorting Binary heas are imlemented using a hea-ordered balanced binary tree. Binomial heas use a collection of B k trees, each of size 2 k. Fibonacci heas use a collection of trees with roerties a bit like B k trees. (The oeration HEAPIFY below makes a hea with n elements without doing n INSERTs.) Binary hea Binomial hea Fibonacci hea Procedure (worst case) (worst case) (amortized) MAKE-HEAP O(1) O(1) O(1) HEAPIFY O(n) O(n) O(n) INSERT O(lg n) O(lg n) O(1) MINIMUM O(1) O(lg n) O(1) EXTRACT-MIN O(lg n) O(lg n) O(lg n) UNION O(n) O(lg n) O(1) DECREASE-KEY O(lg n) O(lg n) O(1) DELETE O(lg n) O(lg n) O(lg n) Binary search trees imlement all the oerations of heas (excet UNION) in addition to SEARCH. Virtually all the oerations take time (log n). The union-nd data structure imlements the oerations UNION(x; y; label) and label FIND(x) on a collection of disjoint sets. Initially (before any UNION oeration) each of n elements is in its own set of size one. A total of m disjoint set oerations take time O(m(m; n)), where (m; n) is a ridiculously slowly growing function. (m; n) is o(log (m)), o(log m), and (1). For comarison-based sorting algorithms, Heasort and mergesort take time O(n log n), and quicksort takes exected time O(n log n). An information theoretic lower bound for any comarison based sort is log(n!) = (n log n). For n numbers known to fall within a range f1; : : : ; Ng, radix sort will take time O(n+N). Linear time algorithms are often ossible if more can be known about the numbers besides how to airwise comare them. Order statistics (the k th largest element of n elements, or the median of n elements) can be found in time (n) by a comarison based algorithm. The algorithm chooses a ivot by recursively comuting the median of the medians of n=5 subsets of 5 elements each. Once all elements are comared to the ivot, 1=4 of the elements can be discarded, as they are all known to be greater than (or, less than) the k th largest element.

2 Initials 2 Exloring grahs Breadth rst search (BFS) takes O(E) time and nds shortest aths. Deth rst search (DFS) takes O(E) time. Also in this time you can have reorder numberings (or discover times), ostorder numberings (or nish times), classication of edges as forward, back, back-cross or back-cross-tree edges. A toological sort of a dag can be found in O(E) time by reversing the ostorder numbers in a DFS. Strongly connected comonents (SCC's) can be found in O(E) time. Note that the comonent grah, G SCC is acyclic, so you can toologically sort it. Minimum Sanning Trees (MST's) If A E is art of a MST, and S is a cut which no edge in A crosses, then the minimum edge across the cut can be added to A to yield art of a MST. Kruskal's grows a collection of trees by always adding the cheaest edge which connects two trees, taking time O(E lg V ) to sort the edges. Prim's algorithm grows a single tree from a vertex by always adding the cheaest edge out from the tree, taking time O(E + V lg V ) if a Fibonacci hea is used. Shortest ath roblems Single-source shortest aths for non-negative edge weights can be found by Dijstra's algorithm. Like Prim's, grow a tree, always adding the vertex with the cheaest ath from the source by extending the tree by only one edge. Takes O(E + V lg V ) using a Fibonacci hea. Single-source shortest aths for edge weights which may be negative can be found using Bellman-Ford. Make V asses over the grah, udating shortest ath estimates to each vertex relaxing edges. Either a negative cycle will be found or a shortest ath tree in time O(EV ). All-airs shortest ath roblems had a few algorithms: Linear rogramming { Matrix-multily like algorithms taking time O(V 4 ) or O(V 3 log V ). { Floyd Warshall takes time time O(V 3 ). They use dynamic rogramming to solve d (k) ij = the shortest ath from v i to v j using aths going through fv 1; : : : ; v kg { Johnson's algorithm rst solves a single source shortest ath roblem from an added vertex, s, (with edges (s; v), v 2 V of weight 0) to reweight the edges by: ^w = w(u; v) + (s; u) (s; v) This results in ositive edge weights, and Bellman-Ford can be used to take time O(EV ). In linear rogramming, the goal is to otimize a linear objective function subject to linear inequality constraints. No algorithm were discussed, but olynomial time algorithms exist for linear rogramming. In integer linear rogramming, the goal is to nd an integer solution to a linear rogramming roblem. olynomial time algorithm is known nor is likely to exist for this NP-comlete variant. Flow networks and maximum ows Caacities satisfy c(u; v) 0. Flows satisfy Caacity constraints : f(u; v) c(u; v) Skew symmetry: f(u; v) = f(v; u) X Flow conservation: 8u 2 V fs; tg : f(u; v) = 0 u2v The residual caacities are given by c f (u; v) = c(u; v) f(u; v). The min-cut max-ow theorem roves the maximum ow is equal to the minimum caacity over all cuts. Further, if there are no augmenting aths in the residual grah, a maximum ow has been obtained. Ford-Fulkerson nds aths from s to t in the residual grah to augment the ows until no more aths can be found, taking time O(Ef ), where f is the value of the max-ow. No

3 Initials 3 Edmonds-Kar imroves on this by always choosing the shortest augmenting ath (i.e., fewest edges), nding the max-ow in time O(V E 2 ). Number theoretic algorithms Throughout, dene to be the length of bits in all the numbers involved. The greatest common divisor gcd(a; b) = gcd(b; a mod b), yielding Euclid's algorithm taking O() arithmetic oerations. If gcd(a; b) = d then there is an x and y so that ax + by = d. Euclid's algorithm can be adjusted to calculated x and y eciently. (Chinese Remainder Theorem) If gcd(n 1; n 2) = 1 and n = n 1n 2, then there is a one-to-one maing between numbers a mod n and airs (a 1modn 1; a 2modn 2) so that a i = a mod n i. To comute a, nd x and y so that n 1x + n 2y = 1 and notice that (1; 0) mas to n 2y mod n and (0; 1) mas to n 1x mod n. So, (a 1; a 2) mas to a 1n 2y + a 2n 1x mod n. (Fermat's Little Theorem) For rime, 1 a <, a 1 1 (mod ). A seudo-rime test is to check if 2 n 1 1 (mod n); outut \rime?" if yes, \comosite!" if no. Very few comosites look like rimes. A randomized rimality test chooses k random values of a in the range 1 a < n. For each, calculate if a n 1 1 (mod n). If one is not 1 outut \comosite!". If all are 1 outut \comosite?". Otherwise outut \rime?". This test fails with robability 1 2 k. In the RSA ublic-key crytosystem, a articiant creates her ublic and rivate keys with the following recedure. 1. Select at random two large rime numbers and q. 2. Comute n by the equations n = q. 3. Select a small odd integer e that is relatively rime to (n) = ( 1)(q 1). 4. Comute d as the multilicative inverse of e mod (n). 5. Publish the air P = (e; n) as her RSA ublic key. 6. Kee secret the air S = (d; n) as her RSA secret key. To encode message M, comuter M e mod n. To decode cybertext C, comute C d mod n

4 Name 4 CS-170 Exam 3 Aril 26, 1995 David Wolfe You should not need to write any code for this exam. Please make your answers as brief and as clear as ossible. I highly recommend crossing out mistakes with a few dark strokes of the en rather than erasing the work comletely in case it is worth artial credit. Below is a summary line for each question on the exam. You may use the backs of ages if you need more sace, but lease indicate for the grader you've done so: For examle, \Continued on back of age 4". Please leave the exam staled. You can look ick u a solution set (with the questions reeated) when you leave. 1. Edmonds-Kar algorithm =20 2. Linear rogramming denitions =15 3. Job scheduling linear rogram =30 4. Non-cycle edges =20 5. Unit-caacity edges =30 Total (Extra oints ossible) =115

5 Name 5 1. (20 oints) The following gives caacities and the ow after executing one round of the Edmonds-Kar imrovement to the Ford-Fulkerson algorithm: j 2=2 - 2=2 js 9 - j @R j @ j j j (a) Draw the residual grah, G f in the sace below. The dotted edges (and the extra grah) are for your convenience. You need not include edges into s or from t in G f. j j sj j j t j j j j j j sj j j t j j j j (b) Draw the ow obtained after one more iteration of Edmonds-Kar. Please do not indicate the caacities; you only need to draw edges with ow. j j sj j j t j j j j j j sj j j t j j j j

6 Name 6 2. (15 oints) Consider the ve roblems labeled (a)-(e) below: min 3x 1 + 4x 2 s.t. x x 2 5 x 2 0 max x 3y + 5z x + y = 4 y + z 2 x + z 6 s.t. min x 1 + x 2 s.t. x 1; x 2 2 f: : : ; 2; 1; 0; 1; 2; 3; : : :g 2x 1 + x 2 5 :5x 1 3x 2 4 (a) (b) (c) min x 1 + x 2 s.t. 2x 1 + x 2 5 :5x 1 3x 2 4 (d) min x 1x 2 s.t. 3x 1 + 2x 2 10 (e) Of the ve, two are linear rograms and one is an integer linear rogram. Indicate which in the boxes below: Linear rogram Linear rogram Integer linear rogram

7 Name 7 3. (30 oints) We want to determine the otimal scheduling of m jobs to a machine such that: All jobs must be comleted within n weeks. Job i requires a total of r i hours. At most h j hours can be scheduled on the machine during week j. There is a cost c ij for each hour that job i is assigned to the machine during week j. This roblem can be formulated as a linear rogram with m n variables, where x ij is the number of hours the machine sends on job i during week j. (a) Write the objective function that minimizes the total cost for a ossible schedule. (b) For job i which requires r i hours, write the corresonding linear constraint. (c) For week j which has at most h j hours available, write the constraint corresonding to the jobs scheduled during this week. (By the way, the additional constraints, \8i; j : x ij 0," will comlete the linear rogram.)

8 Name 8 4. (20 oints) Give an O(E + V ) algorithm to nd all edges in a directed grah G = (V; E) which are not contained in any cycle. Hint: Use deth rst search, bread rst search, strongly connected comonents and/or toological sort as subroutines. (If your algorithm is simle and clearly stated, no justication is required. A one sentence solution could receive full credit.)

9 Name 9 5. (30 oints) Let G = (V; E) be a ow network with source s, sink t, and suose each edge e 2 E has caacity c(e) = 1. Assume also, for convenience, that jej = (V ). (a) Suose we imlement the Ford-Fulkerson maximum-ow algorithm by using deth-rst search to nd augmenting aths in the residual grah. What is the worst case running time of this algorithm on G? (b) Suose a maximum ow for G has been comuted, and a new edge with unit caacity is added to E. Describe how the maximum ow can be eciently udated. (Note: It is not the value of the ow that must be udated, but the ow itself.) Analyze your algorithm. (c) (Extra credit) Suose a maximum ow for G has been comuted, but an edge is now removed from E. Describe how the maximum ow can be eciently udated in O(E + V ) time.

10 Name 10 MORE SPACE IF REQUIRED