Algorithms. Algorithms. Algorithms 6.5 R EDUCTIONS. introduction designing algorithms. establishing lower bounds classifying problems.

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1 Algithms R OBERT S EDGEWICK K EVIN W AYNE Overview: introduction to advanced topics Main topics. [final two lectures] Reduction: relationship between two problems. Algithm design: paradigms f solving problems. 6.5 R EDUCTIONS Shifting gears. introduction designing algithms Algithms F O U R T H E D I T I O N R OBERT S EDGEWICK K EVIN W AYNE establishing lower bounds classifying problems From individual problems to problem-solving models. From linear/quadratic to polynomial/exponential scale. From implementation details to conceptual framewks. Goals. Place algithms and techniques we've studied in a larger context. Introduce you to imptant and essential ideas. Inspire you to learn me about algithms! intractability Last updated on Apr 29, 2015, 5:14 AM 2 Bird's-eye view Desiderata. Classify problems accding to computational requirements. 6.5 R EDUCTIONS introduction complexity der of growth examples linear N min, max, median, Burrows-Wheeler transfm,... linearithmic N log N sting, element distinctness, closest pair, Euclidean MST,... quadratic N2? exponential cn? designing algithms Algithms R OBERT S EDGEWICK K EVIN W AYNE establishing lower bounds classifying problems intractability Frustrating news. Huge number of problems have defied classification. 4

2 Bird's-eye view Reduction Desiderata. Classify problems accding to computational requirements. Desiderata'. Suppose we could (could not) solve problem X efficiently. What else could (could not) we solve efficiently? Def. Problem X reduces to problem Y if you can use an algithm that solves Y to help solve X. (of X) Algithm f Y solution to I Algithm f X Cost of solving X = total cost of solving Y + cost of reduction. perhaps many calls to Y on problems of different sizes (typically only 1 call) preprocessing and postprocessing (typically less than cost of solving Y) Give me a lever long enough and a fulcrum on which to place it, and I shall move the wld. Archimedes 5 6 Reduction Reduction Def. Problem X reduces to problem Y if you can use an algithm that solves Y to help solve X. Def. Problem X reduces to problem Y if you can use an algithm that solves Y to help solve X. (of X) Algithm f Y solution to I (of X) Algithm f Y solution to I Algithm f X Algithm f X Ex 1. [finding the median reduces to sting] To find the median of N items: St N items. Return item in the middle. cost of sting Cost of finding the median. N log N + 1. cost of reduction Ex 2. [element distinctness reduces to sting] To solve element distinctness on N items: St N items. Check adjacent pairs f equality. cost of sting Cost of element distinctness. N log N + N. cost of reduction 7 8

3 Reduction Reductions: quiz 1 Def. Problem X reduces to problem Y if you can use an algithm that solves Y to help solve X. Which of the following reductions have we encountered in this course? I. MAX-FLOW reduces to MIN-CUT. II. MIN-CUT reduces to MAX-FLOW. need to find max st-flow and min st-cut (not simply compute the value) (of X) Algithm f Y solution to I A. I only. B. II only. Algithm f X C. Both I and II. Novice err. Confusing X reduces to Y with Y reduces to X. D. Neither I n II. E. I don't know. ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHEIDUNGSPROBLEM By A. M. TURING. [Received 28 May, Read 12 November, 1936.] 9 10 Reduction: design algithms Def. Problem X reduces to problem Y if you can use an algithm that solves Y to help solve X. Algithms ROBERT SEDGEWICK KEVIN WAYNE REDUCTIONS introduction designing algithms establishing lower bounds classifying problems intractability Design algithm. Given an algithm f Y, can also solve X. Me familiar reductions. Mincut reduces to maxflow. Arbitrage reduces to negative cycles. Bipartite matching reduces to maxflow. Seam carving reduces to shtest paths in a DAG. Burrows-Wheeler transfm reduces to suffix st. Mentality. Since I know how to solve Y, can I use that algithm to solve X? programmer s version: I have code f Y. Can I use it f X? 12

4 3-collinear 3-collinear reduces to sting 3-COLLINEAR. Given N distinct points in the plane, are there 3 ( me) that all lie on the same line? Sting-based algithm. F each point p, Compute the slope that each other point q makes with p. St the N 1 points by slope. Collinear points are adjacent. q 4 q 3 q 2 q 1 dy1 p dx1 3-collinear Brute fce N 3. F all triples of points (p, q, r), check if they are collinear. cost of sting (N times) Cost of solving 3-COLLINEAR. N 2 log N + N 2. cost of reduction Shtest paths on edge-weighted graphs and digraphs Some reductions in combinatial optimization Proposition. Undirected shtest paths (with nonnegative weights) reduces to directed shtest path baseball elimination mincut bipartite matching undirected shtest paths (nonnegative) seam carving s t maxflow directed shtest paths (nonnegative) arbitrage shtest paths (in a DAG) Pf. Replace each undirected edge by two directed edges assignment problem directed shtest paths (no neg cycles) s t Cost of solving undirected shtest paths. E log V + (E + V). linear programming cost of Dijkstra cost of reduction 15 16

5 Bird's-eye view Goal. Prove that a problem requires a certain number of steps. Ex. In decision tree model, any compare-based sting algithm requires Ω(N log N) compares in the wst case. 6.5 REDUCTIONS yes a < b no Algithms ROBERT SEDGEWICK KEVIN WAYNE introduction designing algithms establishing lower bounds classifying problems intractability yes a b c b < c no a < c yes a c b no c a b yes b a c a < c yes b c a no b < c no c b a argument must apply to all conceivable algithms Bad news. Very difficult to establish lower bounds from scratch. Good news. Spread Ω(N log N) lower bound to Y by reducing sting to Y. assuming cost of reduction is not too high 18 Linear-time reductions Reductions: quiz 2 Def. Problem X linear-time reduces to problem Y if X can be solved with: Linear number of standard computational steps. Constant number of calls to Y. Establish lower bound: If X takes Ω(N log N) steps, then so does Y. If X takes Ω(N 2 ) steps, then so does Y. Mentality. If I could easily solve Y, then I could easily solve X. I can t easily solve X. Therefe, I can't easily solve Y. Which of the following reductions is not a linear-time reduction? A. ELEMENT-DISTINCTNESS reduces to SORTING. B. MIN-CUT reduces to MAX-FLOW. C. 3-COLLINEAR reduces to SORTING. D. BURROWS-WHEELER-TRANSFORM reduces to SUFFIX-SORTING. E. I don't know

6 ELEMENT-DISTINCTNESS linear-time reduces to 2D-CLOSEST-PAIR ELEMENT-DISTINCTNESS linear-time reduces to 2D-CLOSEST-PAIR ELEMENT-DISTINCTNESS. Given N elements, are any two equal? 2D-CLOSEST-PAIR. Given N points in the plane, find the closest pair. ELEMENT-DISTINCTNESS. Given N elements, are any two equal? 2D-CLOSEST-PAIR. Given N points in the plane, find the closest pair Proposition. ELEMENT-DISTINCTNESS linear-time reduces to 2D-CLOSEST-PAIR. Pf. ELEMENT-DISTINCTNESS instance: x1, x2,..., xn. 2D-CLOSEST-PAIR instance: (x1, x1), (x2, x2),..., (xn, xn). The N elements are distinct iff distance of closest pair > 0. allows linear tests like xi < xj and quadratic tests like (x i x k) 2 + (x j x k) 2 > 4 ELEMENT-DISTINCTNESS lower bound. In quadratic decision tree model, any algithm that solves ELEMENT-DISTINCTNESS takes Ω(N log N) steps. element distinctness 2d closest pair Implication. In quadratic decision tree model, any algithm f 2D-CLOSEST-PAIR takes Ω(N log N) steps Some linear-time reductions in computational geometry Lower bound f 3-COLLINEAR 3-SUM. Given N distinct integers, are there three that sum to 0? element distinctness (N log N lower bound) 3-COLLINEAR. Given N distinct points in the plane, are there 3 ( me) that lie on the same line? sting 2d convex hull 2d closest pair 2d Euclidean MST smallest enclosing circle Delaunay triangulation Vonoi diagram largest empty circle (N log N lower bound) 3-sum 3-collinear 23 24

7 Lower bound f 3-COLLINEAR 3-SUM linear-time reduces to 3-COLLINEAR 3-SUM. Given N distinct integers, are there three that sum to 0? 3-COLLINEAR. Given N distinct points in the plane, are there 3 ( me) that lie on the same line? Proposition. 3-SUM linear-time reduces to 3-COLLINEAR. 3-SUM instance: x1, x2,..., xn. 3-COLLINEAR instance: (x1, x13 ), (x2, x2 3 ),..., (xn, xn 3 ). Lemma. If a, b, and c are distinct, then a + b + c = 0 if and only if (a, a 3 ), (b, b 3 ), and (c, c 3 ) are collinear. f (x) = x 3 Proposition. 3-SUM linear-time reduces to 3-COLLINEAR. Pf. [next two slides] lower-bound mentality: if I can't solve 3-SUM in N 1.99 time, I can't solve 3-COLLINEAR in N 1.99 time either (1, 1) (2, 8) Conjecture. Any algithm f 3-SUM requires Ω(N 2 ε ) steps. Implication. No sub-quadratic algithm f 3-COLLINEAR likely. our N 2 log N algithm was pretty good (-3, -27) = SUM linear-time reduces to 3-COLLINEAR Me geometric reductions and lower bounds Proposition. 3-SUM linear-time reduces to 3-COLLINEAR. 3-SUM instance: x1, x2,..., xn. 3-COLLINEAR instance: (x1, x13 ), (x2, x2 3 ),..., (xn, xn 3 ). Lemma. If a, b, and c are distinct, then a + b + c = 0 if and only if (a, a 3 ), (b, b 3 ), and (c, c 3 ) are collinear. 3-SUM (conjectured N 2 ε lower bound) Pf. Three distinct points (a, a 3 ), (b, b 3 ), and (c, c 3 ) are collinear iff: POLYGONAL-CONTAINMENT 3-COLLINEAR DIHEDRAL-ROTATION GEOMETRIC-BASE 0 = a a 3 1 b b 3 1 c c 3 1 = a(b 3 c 3 ) b(a 3 c 3 )+c(a 3 b 3 ) 3-CONCURRENT MIN-AREA-TRIANGLE SEPARATOR LINE-SEGMENT- PLANAR-MOTION- PLANNING = (a b)(b c)(c a)(a + b + c) 27 28

8 Complexity of 3-SUM Establishing lower bounds: summary April Some recent evidence that the complexity might be N 3 / 2. Establishing lower bounds through reduction is an imptant tool in guiding algithm design effts. Q. How to convince yourself no linear-time EUCLIDEAN-MST algithm exists? A1. [hard way] Long futile search f a linear-time algithm. A2. [easy way] Linear-time reduction from element distinctness. Threesomes, Degenerates, and Love Triangles Allan Grønlund MADALGO, Aarhus University Seth Pettie University of Michigan arxiv: v1 [cs.ds] 3 Apr 2014 April 4, 2014 Abstract The 3SUM problem is to decide, given a set of n real numbers,? whether any three sum to zero. We prove that the decision tree complexity of 3SUM is Opn3{2 log nq, that there is a randomized 3SUM algithm running in Opn2 plog log nq2 { log nq time, and a deterministic algithm running in Opn2 plog log nq5{3 {plog nq2{3 q time. These results refute the strongest version of the 3SUM conjecture, namely that its decision tree (and algithmic) complexity is Ωpn2 q. Our results lead directly to improved algithms f k-variate linear degeneracy testing f all ř odd k ě 3. The problem is to decide, given a linear function f px1,..., xk q α0 ` 1ďiďk αi xi? and a set S Ă R, whether 0 P f ps k q. We show the decision tree complexity is Opnk{2 log nq and give algithms running in time Opnpk`1q{2 { polyplog nqq. Finally, we give a subcubic algithm f a generalization of the pmin, `q-product over realvalued matrices? and apply it to the problem of finding zero-weight triangles in weighted graphs. A depth-opn5{2 log nq decision tree is given f this problem, as well as an algithm running in time Opn3 plog log nq2 { log nq. 2d Euclidean MST Introduction The time hierarchy theem [16] implies that there exist problems in P with complexity Ωpnk q f every fixed k. However, it is consistent with current knowledge that all problems of practical interest can be solved in O pnq time in a reasonable model of computation. Effts to build a useful complexity they inside P have been based on the conjectured hardness of certain archetypal problems, such as 3SUM, pmin, `q-matrix product, and CNF-SAT. See, f example, the conditional lower bounds in [15, 19, 20, 17, 1, 2, 21, 10, 23]. In this paper we study the complexity of 3SUM and related problems such as linear degeneracy testing (LDT) and finding zero-weight triangles. Let us define the problems fmally. 3SUM: Given a set S Ă R, determine if there exists a, b, c P S such that a ` b ` c 0. Classifying problems: summary Desiderata. Problem with algithm that matches lower bound. Ex. Sting and element distinctness have complexity N log N. Integer3SUM: Given a set S Ď t U,..., U u Ă Z, determine if there exists a, b, c P S such that a ` b ` c 0. This wk is suppted in part by the Danish National Research Foundation grant DNRF84 through the Center f Massive Data Algithmics (MADALGO). S. Pettie is suppted by NSF grants CCF and CNS and a grant from the US-Israel Binational Science Foundation. 6.5 R EDUCTIONS Desiderata'. Prove that two problems X and Y have the same complexity. First, show that problem X linear-time reduces to Y. 1 introduction designing algithms Algithms R OBERT S EDGEWICK K EVIN W AYNE Second, show that Y linear-time reduces to X. Conclude that X has complexity N iff Y has complexity N b establishing lower bounds b f b 1. even if we don't know what it is classifying problems intractability X = sting Y = element distinctness integer multiplication integer division 32

9 Integer arithmetic reductions Integer arithmetic reductions Integer multiplication. Given two N-bit integers, compute their product. Brute fce. N 2 bit operations. Integer multiplication. Given two N-bit integers, compute their product. Brute fce. N 2 bit operations problem arithmetic der of growth integer multiplication a b M(N) integer division a / b, a mod b M(N) integer square a 2 M(N) integer square root a M(N) integer arithmetic problems with the same complexity as integer multiplication Q. Is brute-fce algithm optimal? Histy of complexity of integer multiplication Numerical linear algebra reductions year algithm der of growth? brute fce N 2 Matrix multiplication. Given two N-by-N matrices, compute their product. Brute fce. N 3 flops Karatsuba N Toom-3, Toom-4 N 1.465, N column j j 1966 Toom-Cook N 1 + ε Schönhage Strassen N log N log log N 2007 Fürer N log N 2 log*n row i = i ?? N number of bit operations to multiply two N-bit integers used in Maple, Mathematica, gcc, cryptography, = 0.47 Remark. GNU Multiple Precision Library uses one of five different algithm depending on size of operands

10 Numerical linear algebra reductions Histy of complexity of matrix multiplication Matrix multiplication. Given two N-by-N matrices, compute their product. Brute fce. N 3 flops. problem linear algebra der of growth matrix multiplication A B MM(N) matrix inversion A 1 MM(N) determinant A MM(N) system of linear equations Ax = b MM(N) LU decomposition A = L U MM(N) least squares min Ax b 2 MM(N) numerical linear algebra problems with the same complexity as matrix multiplication year algithm der of growth? brute fce N Strassen N Pan N Bini N Schönhage N Romani N Coppersmith-Winograd N Strassen N Coppersmith-Winograd N Strother N Williams N de Gall N Q. Is brute-fce algithm optimal? 37?? N 2 + ε number of floating-point operations to multiply two N-by-N matrices 38 Bird's-eye view Def. A problem is intractable if it can't be solved in polynomial time. Desiderata. Prove that a problem is intractable. Algithms ROBERT SEDGEWICK KEVIN WAYNE REDUCTIONS introduction designing algithms establishing lower bounds classifying problems intractability Two problems that provably require exponential time. input size = c + lg K Given a constant-size program, does it halt in at most K steps? Given N-by-N checkers board position, can the first player fce a win? using fced capture rule Frustrating news. Very few successes. 40

11 A ce problem: satisfiability Satisfiability is conjectured to be intractable SAT. Given a system of boolean equations, find a solution. Ex. x 1 x 2 x 3 = true x 1 x 2 x 3 = true x 1 x 2 x 3 = true x 1 x 2 x 4 = true x 2 x 3 x 4 = true x1 x2 x3 x4 T T F T Q. How to solve an instance of 3-SAT with N variables? A. Exhaustive search: try all 2 N truth assignments. solution S 3-SAT. All equations of this fm (with three variables per equation). Key applications. Automatic verification systems f software. Mean field diluted spin glass model in physics. Electronic design automation (EDA) f hardware Q. Can we do anything substantially me clever? Conjecture (P NP). 3-SAT is intractable (no poly-time algithm). consensus opinion 42 Polynomial-time reductions Integer linear programming Problem X poly-time (Cook) reduces to problem Y if X can be solved with: Polynomial number of standard computational steps. Polynomial number of calls to Y. ILP. Given a system of linear inequalities, find an integral solution. 3x1 + 5x2 + 2x3 + x4 + 4x5 10 5x1 + 2x2 + 4x4 + 1x5 7 (of X) Algithm f Y solution to I x1 + x3 + 2x4 2 3x1 + 4x3 + 7x4 7 linear inequalities Algithm f X x1 + x4 1 Establish intractability. If 3-SAT poly-time reduces to Y, then Y is intractable. (assuming 3-SAT is intractable) x1 + x3 + x5 1 all xi = { 0, 1 } integer variables x1 x2 x3 x4 x Mentality. If I could solve Y in poly-time, then I could also solve 3-SAT in poly-time. 3-SAT is believed to be intractable. Therefe, so is Y. solution S Context. Cnerstone problem in operations research. Remark. Finding a real-valued solution is tractable (linear programming)

12 3-SAT poly-time reduces to ILP Reductions: quiz 3 3-SAT. Given a system of boolean equations, find a solution. x 1 x 2 x 3 = true x 1 x 2 x 3 = true x 1 x 2 x 3 = true x 1 x 2 x 4 = true x 2 x 3 x 4 = true ILP. Given a system of linear inequalities, find a 0-1 solution. (1 x 1 ) + x 2 + x 3 1 Suppose that Problem X poly-time reduces to Problem Y. Which of the following can you infer? A. If X can be solved in poly-time, then so can Y. B. If X cannot be solved in cubic time, Y cannot be solved in poly-time. C. If Y can be solved in cubic time, then X can be solved in poly-time. D. If Y cannot be solved in poly-time, then neither can X. E. I don't know. x 1 + (1 x 2 ) + x 3 1 (1 x 1 ) + (1 x 2 ) + (1 x 3 ) 1 (1 x 1 ) + (1 x 2 ) + + x 4 1 (1 x 2 ) + x 3 + x 4 1 solution to this ILP instance gives solution to iginal 3-SAT instance Me poly-time reductions from 3-satisfiability Implications of poly-time reductions from 3-satisfiability 3-SAT Establishing intractability through poly-time reduction is an imptant tool in guiding algithm design effts. 3-COLOR 3-SAT reduces to ILP VERTEX-COVER Dick Karp '85 Turing award Q. How to convince yourself that a new problem is (probably) intractable? A1. [hard way] Long futile search f an efficient algithm (as f 3-SAT). A2. [easy way] Reduction from 3-SAT. EXACT-COVER ILP CLIQUE HAM-CYCLE Caveat. Intricate reductions are common. SUBSET-SUM TSP HAM-PATH PARTITION KNAPSACK BIN-PACKING Conjecture. 3-SAT is intractable. Implication. All of these problems are intractable

13 Search problems P vs. NP Search problem. Problem where you can check a solution in poly-time. P. Set of search problems solvable in poly-time. Imptance. What scientists and engineers can compute feasibly. Ex 1. 3-SAT. x1 x2 x3 = true x1 x2 x3 = true x1 x2 x3 = true x1 x2 x2 x3 x4 = true x4 = true NP. Set of search problems (checkable in poly-time). Imptance. What scientists and engineers aspire to compute feasibly. Fundamental question. x1 x2 x3 x4 T T F T solution S Ex 2. FACTOR. Given an N-bit integer x, find a nontrivial fact solution S Consensus opinion. No. 49 Cook-Levin theem 50 Implications of Cook-Levin theem A problem is NP-COMPLETE if 3-SAT It is in NP. All problems in NP poly-time to reduce to it. T R OLO u red ces -SA to 3 3-C IND-SET 3-COLOR VERTEX-COVER Cook-Levin theem. 3-SAT is NP-COMPLETE. Stephen Cook '82 Turing award Collary. 3-SAT is tractable if and only if P = NP. Leonid Levin ILP CLIQUE EXACT-COVER HAM-CYCLE Two wlds. SUBSET-SUM NP P NPC TSP HAM-PATH P = NP PARTITION All of these problems (and many, many me) P NP P = NP KNAPSACK 51 BIN-PACKING poly-time reduce to 3-SAT. 52

14 Implications of Karp + Cook-Levin Reductions: quiz 4 3-COLOR 3-COLOR reduces to 3-SAT 3-SAT reduces to 3-COLOR 3-SAT IND-SET VERTEX-COVER + Suppose that X is NP-COMPLETE, Y is in NP, and X poly-time reduces to Y. Which of the following statements can you infer? I. Y is NP-COMPLETE. II. If Y cannot be solved in poly-time, then P NP. ILP III. If P NP, then neither X n Y can be solved in poly-time. EXACT-COVER CLIQUE HAM-CYCLE A. I only. B. II only. SUBSET-SUM TSP HAM-PATH C. I and II only. D. I, II, and III. PARTITION E. I don't know. KNAPSACK BIN-PACKING All of these problems are NP-COMPLETE; they are manifestations of the same really hard problem Birds-eye view: review Birds-eye view: revised Desiderata. Classify problems accding to computational requirements. Desiderata. Classify problems accding to computational requirements. complexity der of growth examples complexity der of growth examples linear N min, max, median, Burrows-Wheeler transfm,... linear N min, max, median, Burrows-Wheeler transfm,... linearithmic N log N sting, element distinctness,... linearithmic N log N sting, element distinctness,... quadratic N 2? M(N)? MM(N)? integer multiplication, division, square root,... matrix multiplication, Ax = b, least square, determinant,... exponential c N? NP-complete probably not N b 3-SAT, IND-SET, ILP,... Frustrating news. Huge number of problems have defied classification. Good news. Can put many problems into equivalence classes

15 Complexity zoo Summary Complexity class. Set of problems sharing some computational property. Reductions are imptant in they to: Design algithms. Establish lower bounds. Classify problems accding to their computational requirements. Text Reductions are imptant in practice to: Design algithms. Design reusable software modules. stacks, queues, priity queues, symbol tables, sets, graphs sting, regular expressions, suffix arrays MST, shtest paths, maxflow, linear programming Determine difficulty of your problem and choose the right tool. Bad news. Lots of complexity classes (496 animals in zoo)