Algorithms q AlKhwarizmi arab mathematician 8 th century

Algorithms q. Al-Khwarizmi, arab mathematician, 8 th century q. Wrote a book: al-kitab… from which the word Algebra comes q. Oldest algorithm: Euclidian algorithm for finding the largest common divisor of two numbers

Ten Big Ideas in Algorithms 1. Orders of Magnitude and Big Oh notation 2. Complexity: Polynomial versus Exponential 3. Iteration versus Recursion 4. Recurrence Relations 5. Divide-and-Conquer 6. Graph Traversal: Depth-First, Breadth-First 7. Greedy Algorithm 8. Dynamic Programming 9. Lower Bounds 10. NP-completeness

1. Orders of Magnitude • How to compare algorithm A with algorithm B in an implementation-independent manner? • Asymptotic analysis: compare them when n goes to infinity (becomes very large) • [An aside: massive data sets; the Internet and search engines; Bioinformatics] • a(n)=running time of alg A, b(n)=of alg B • A is better than B if lim a(n)/b(n) = 0 (b “grows faster” than b) • Examples in Mathematica

Big Oh Notation • Too many functions, not all components relevant • Big Oh helps keep the big picture. • Example: a(n) = 3 n 2 + 1 and b(n) = 5 n 2 + 3 n + 100 grow roughly at the same rate • lim a(n) / b(n) = constant • a(n) = O(n 2), b(n) = O(n 2)

Complexity: Polynomial versus Exponential • Polynomial: order growth of O(n), O(n 2), O(n 3), etc. is tractable. • Exponential: order growth of O(2 n), O(nn) or larger is intractable and leads to impractical algorithms

3. Iteration versus Recursion • Find Min of n elements: iterate over the array, at each step do a comparison. Time = n • Sort n elements by Selection Sort: iterate over array, at each iteration Find Min, put in right position Time = n + (n-1) + (n-2) + … + 3 + 2 + 1 • Find closed form for summation: Time = n(n+1)/2= O(n 2)

Recursion • Binary Search: split the list in two, search in the half where it should be. • Merge Sort: divide the array in two equal parts; sort each part recursively; merge them. • Quick Sort: use first element to partition array in two, first being smaller than second; sort each part recursively.

4. Recurrence Relations • Time for Merge Sort: F(n) = 2 F(n/2) + n • Solve recurrence relation: F(n) = n log n

5. Divide and Conquer • Paradigm for algorithm design: split in two or more parts, solve recursively, combine solutions • Binary Search, Merge Sort and Quick Sort illustrate it. • Leads to improved algorithms for many problems: o Median finding o Matrix Multiplication o Computational Geometry algorithms (convex hulls, etc. ) • Leads to divide and conquer recurrence relations.

6. Graph Traversal: Depth-First and Breadth-First • Techniques for designing very efficient (linear time) algorithms • Apply to a variety of graph problems: • • • Connectivity Finding cycles in graphs Strongly connected components in directed graphs Biconnected components in graphs Planarity testing

7. Greedy Algorithm • Proceed iteratively. At each step, choose an element which maximizes some “profit” or minimizes some “cost”. • Minimum spanning tree in a graph (min cost connecting network): at each step, pick up the least expensive edge that does not create a cycle in the graph. • Shortest paths in a graph from given vertex a: at each step, choose a new vertex which is closest to an already constructed set of vertices.

8. Dynamic Programming • A scheme for combining several results for smaller values to obtain the current result. • Often leads to polynomial time algorithms • All pairs shortest paths: compute the shortest path between all pairs of nodes in a graph.

9. Lower Bounds • When do you stop improving an algorithm? How to prove that what you found is the best possible algorithm? • Lower bounds: techniques for proving an algorithm is the best possible. • Work under certain assumptions about the model of computation • Classical bounds: finding the minimum, sorting. • Information Theoretical Lower Bound • Adversary arguments

10. NP-Completeness • Despite all efforts, hundreds of very practical problems have no efficient (polynomial time) known algorithm (doesn’t mean that one may not be found) • Nobody has been able to prove that no polynomial time algorithm exists for any of these problems. • But one can show that if a polynomial time algorithm exists for one of these problems, then it exists for all of them. • A problem is NP-complete if it is as hard (NPhard) as any of the problems in this class, and if a solution to the problem can be verified in polynomial time (it is in NP)

Some NP-complete Problems • • Traveling Salesman Problem Hamiltonian Cycle in a graph Maximum Clique in a graph Satisfiability for boolean formulas and circuits • The game of Push-Push

Overview of the course • • • We cover the 10 big ideas Lots of examples of algorithms Start with some algebra and calculus Need data structures and recursion Use Leda software for demos (all algorithms are implemented and visualized) • Abstract, high level class • Need to be comfortable with abstract thinking • A little bit of programming – mostly theory