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Dept of CS & IT University of SargodhaAdvance Analysis of Algorithms Fall 2011 Fahad Maqbool MS Computer Science BS Software Engineering 1
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Adv Analysis of AlgoSteps involved in writing a computer program: • Problem formulation & specification • Design of solution • Implementation • Testing • Documentation • Evaluation of the solution 2
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Adv Analysis of AlgoProblem formulation & specification• Half the battle is knowing what problem to solve.• Most problems have no simple precise specification.• Some problems are impossible to formulate.• Example: Gourmet recipe, World peace, Etc. 3
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Adv Analysis of AlgoHow to formulate & specify a problem?• Identify the problem parameters.• Expressing the problem by a formal model.• Looking for a solution for the model.• In the absence of a solution, discover about the model. 4
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Adv Analysis of AlgoHow to model and solve a problem?• Using any branch of Mathematics or Science.• Identifying problems of numerical nature.• With a suitable model, attempt to find a solution in terms of thatmodel• Simultaneous linear equations for electrical circuits.• Differential equations for predicting chemical reaction rates. 5
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Adv Analysis of Algo Algorithm • Finite sequence of instructions. • Each instruction having a clear meaning. • Each instruction requiring finite amount of effort. • Each instruction requiring finite time to complete. 6
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Adv Analysis of Algo Algorithm • Finite sequence of instructions. An input should not take the program in an infinite loop • Each instruction having a clear meaning. Very subjective. What is clear to me, may not be clear to you. • Each instruction requiring finite amount of effort. Very subjective. Finite on a super computer or a 486? • Each instruction requiring finite time to complete. Very subjective. 1 min, 1 hr, 1 year or a lifetime? 7
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Adv Analysis of AlgoUsing a model to solve a complicated traffic lightproblemGIVEN: A complex intersection.OBJECTIVE: Traffic light with minimum phases.SOLUTION:• Identify permitted turns, going straight is a “turn”.• Make group of permitted turns.• Make the smallest possible number of groups.• Assign each phase of the traffic light to a group. 8
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Adv Analysis of AlgoUsing a model to solve a complicated traffic lightproblem D E C Rawalpindi B A An intersection Roads C and E are one way, others are two way. 9
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Adv Analysis of AlgoUsing a model to solve a complicated traffic lightproblemThere are 13 permitted turns.Some turns such as AB (from A to B) and EC can be carriedout simultaneously.Other like AD and EB cross each other and can not be carriedout simultaneously.The traffic light should permit AB and EC simultaneously, butshould not allow AD and EB. 10
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Adv Analysis of AlgoUsing a model to solve a complicated traffic lightproblem D AB & AC E C AD & EB B A An intersection 11
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Adv Analysis of AlgoUsing a model to solve a complicated traffic lightproblemSOLUTION:• We model the problem using a structure called graphG(V,E).• A graph consists of a set of points called vertices V, and linesconnecting the points, called edges E.•Drawing a graph such that the vertices represent turns.• Edges between those turns that can NOT be performedsimultaneously. 12
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Adv Analysis of AlgoUsing a model to solve a complicated traffic lightproblem D E AB AC AD C BA BC BD B DA DB DC A An intersection EA EB EC ED Partial graph of incompatible turns 13
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Adv Analysis of AlgoUsing a model to solve a complicated traffic lightproblem Partial table of incompatible turns 14
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Adv Analysis of AlgoUsing a model to solve a complicated traffic lightproblemSOLUTION:The graph can aid in solving our problem. A coloring of a graph is an assignment of a color to eachvertex of the graph, so that no two vertices connected by anedge have the same color.Our problem is of coloring the graph of incompatible turnsusing as few colors as possible. 15
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Adv Analysis of AlgoUsing a model to solve a complicated traffic lightproblemMore on SOLUTION (Graph coloring):Graph coloring problem has been studied for decades.The theory of algorithms tells us a lot about it.Unfortunately this belongs to a class of problems called asNP-Complete problems.For such problems, all known solutions are basically “try allpossibilities”In case of coloring, try all assignments of colors. 16
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Adv Analysis of AlgoUsing a model to solve a complicated traffic lightproblemApproaches to attempting NP-Complete problems:1. If the problem is small, might attempt to find an optimal solution exhaustively.2. Look for additional information about the problem.3. Change the problem a little, and look for a good, but not necessarily optimal solution. An algorithm that quickly produces good but not necessarily optimal solutions is called a heuristic. 17
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Using model to solve complicated traffic light problem A reasonable heuristic for graph coloring is the greedy “algorithm”. Try to color as many vertices as possible with the first color, and then as many uncolored vertices with the second color, and so on. The approach would be:1. Select some uncolored vertex, and color with new color.2. Scan the list of uncolored vertices.3. For each uncolored vertex, determine whether it has an edge to any vertex already colored with the new color.4. If there is no such edge, color the present vertices with the new color. 18
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Using model to solve complicated traffic light problem This is called “greedy”, because it colors a vertex, whenever it can, without considering potential drawbacks. 3 1 5 2 Three colors used 3 4 1 5 2 Two colors used (optimum) 4 19
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Analysis of AlgoUsing a model to solve a complicated traffic lightproblem Work SMART not HARD 20
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Questions that will be answered• What is a “good” or "efficient" program?• How to measure the efficiency of a program?• How to analyse a simple program?• How to compare different programs?• What is the big-O notation?• What is the impact of input on program performance?• What are the standard program analysis techniques?• Do we need fast machines or fast algorithms? 21/31
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Which is better?The running time of a program. Program easy to understand? Program easy to code and debug? Program making efficient use of resources? Program running as fast as possible? 22/31
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Measuring EfficiencyWays of measuring efficiency:• Run the program and see how long it takes• Run the program and see how much memory it uses•Lots of variables to control:• What is the input data?• What is the hardware platform?• What is the programming language/compiler?• Just because one program is faster than another right now, means itwill always be faster? 23/31
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Measuring EfficiencyWant to achieve platform-independence• Use an abstract machine that uses steps of time and units of memory, instead of seconds or bytes• - each elementary operation takes 1 step• - each elementary instance occupies 1 unit of memory 24/31
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A Simple Example// Input: int A[N], array of N integers// Output: Sum of all numbers in array Aint Sum(int A[], int N) { int s=0; for (int i=0; i< N; i++) s = s + A[i]; return s;}How should we analyse this? 25/31
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A Simple Example Analysis of Sum 1.) Describe the size of the input in terms of one ore more parameters: - Input to Sum is an array of N ints, so size is N. 2.) Then, count how many steps are used for an input of that size: - A step is an elementary operation such as +, <, =, A[i] 26/31
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A Simple Example Analysis of Sum (2) // Input: int A[N], array of N integers // Output: Sum of all numbers in array A int Sum(int A[], int N { int s=0; 1 for (int i=0; i< N; i++) 2 3 4 s = s + A[i]; 5 6 7 1,2,8: Once return s; 3,4,5,6,7: Once per each iteration } 8 of for loop, N iteration Total: 5N + 3 The complexity function of the algorithm is : f(N) = 5N +3 27/31
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Analysis: A Simple Example How 5N+3 Grows Estimated running time for different values of N: N = 10 => 53 steps N = 100 => 503 steps N = 1,000 => 5003 steps N = 1,000,000 => 5,000,003 steps As N grows, the number of steps grow in linear proportion to N for this Sum function. 28/31
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Analysis: A Simple Example What Dominates? What about the 5 in 5N+3? What about the +3? • As N gets large, the +3 becomes insignificant •5 is inaccurate, as different operations require varying amounts of time What is fundamental is that the time is linear in N. Asymptotic Complexity: As N gets large, concentrate on the highest order term: • Drop lower order terms such as +3 • Drop the constant coefficient of the highest order term i.e. N 29/31
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Analysis: A Simple Example Asymptotic Complexity • The 5N+3 time bound is said to "grow asymptotically" like N • This gives us an approximation of the complexity of the algorithm • Ignores lots of (machine dependent) details, concentrate on the bigger picture 30/31
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Comparing FunctionsDefinition: If f(N) and g(N) are two complexity functions, we say f(N) = O(g(N))(read "f(N) as order g(N)", or "f(N) is big-O of g(N)")if there are constants c and N0 such that for N N0, T(N) cN f(N) c g(N)for all sufficiently large N. 31/31
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Comparing Functions 100n2 Vs 5n3, which one is better? Differenec of functions 20000 10000 0-10000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33-20000-30000-40000-50000-60000-70000-80000-90000 33/31
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Comparing Functions Why is this useful? As inputs get larger, any algorithm of a smaller order will be more efficient than an algorithm of a larger order 0.05 N2 = O(N2) Time (steps) 3N = O(N) Input (size) N = 60 34/31
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Comparing FunctionsWhat is the relationship between the following?What is the polynomial representation for the following? f1(n) f2(n) f3(n) F(n) f4(n) Input size (n) 35/31
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Big-O Notation• Think of f(N) = O(g(N)) as " f(N) grows at most like g(N)" or " f grows no faster than g" (ignoring constant factors, and for large N)Important:• Big-O is not a function!• Never read = as "equals"• Examples: 5N + 3 = O(N) 37N5 + 7N2 - 2N + 1 = O(N5) 36/31
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Size does matter Common Orders of Growth Let N be the input size, and b and k be constants O(k) = O(1) Constant Time Increasing Complexity O(logbN) = O(log N) Logarithmic Time O(N) Linear Time O(N log N) O(N2) Quadratic Time O(N3) Cubic Time ... O(kN) Exponential Time 38/31
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Size does matterWhat happens if we double the input size N? N log2N 5N N log2N N2 2N 8 3 40 24 64 256 16 4 80 64 256 65536 32 5 160 160 1024 ~109 64 6 320 384 4096 ~1019128 7 640 896 16384 ~1038256 8 1280 2048 65536 ~1076 If one execution takes 10-6 seconds, 2256 will take about 1061 centuries 39/31
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Size does matter Big NumbersSuppose a program has run time O(n!) and the run timefor n = 10 is 1 secondFor n = 12, the run time is 2 minutesFor n = 14, the run time is 6 hoursFor n = 16, the run time is 2 monthsFor n = 18, the run time is 50 yearsFor n = 20, the run time is 200 centuries 40/31
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Standard Analysis TechniquesConstant time statements Simplest case: O(1) time statements • Assignment statements of simple data types int x = y; • Arithmetic operations: x = 5 * y + 4 - z; • Array referencing: A[j] = 5; • Array assignment: j, A[j] = 5; • Most conditional tests: if (x < 12) ... 41/31
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Standard Analysis Techniques Analyzing Loops Any loop has two parts: 1. How many iterations are performed? 2. How many steps per iteration? int sum = 0,j; for (j=0; j < N; j++) sum = sum +j; - Loop executes N times (0..N-1) - 4 = O(1) steps per iteration - Total time is N * O(1) = O(N*1) = O(N) 42/31
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Standard Analysis Techniques Analyzing Loops (2) What about this for-loop? int sum =0, j; for (j=0; j < 100; j++) sum = sum +j; - Loop executes 100 times - 4 = O(1) steps per iteration - Total time is 100 * O(1) = O(100 * 1) = O(100) = O(1) 43/31
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Standard Analysis Techniques Analyzing Loops (3) What about while-loops? Determine how many times the loop will be executed: bool done = false; int result = 1, n; scanf("%d", &n); while (!done){ result = result *n; n--; if (n <= 1) done = true; } Loop terminates when done == true, which happens after N iterations. Total time: O(N) 44/31
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Standard Analysis Techniques Nested Loops Treat just like a single loop and evaluate each level of nesting as needed: int j,k; for (j=0; j<N; j++) for (k=N; k>0; k--) sum += k+j; Start with outer loop: - How many iterations? N - How much time per iteration? Need to evaluate inner loop Inner loop uses O(N) time Total time is N * O(N) = O(N*N) = O(N2) PRODUCT RULE 45/31
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Standard Analysis Techniques Nested Loops (2) What if the number of iterations of one loop depends on the counter of the other? int j,k; for (j=0; j < N; j++) for (k=0; k < j; k++) sum += k+j; Analyze inner and outer loop together: - Number of iterations of the inner loop is: 0 + 1 + 2 + ... + (N-1) = O(N2) How? 46/31
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Standard Analysis Techniques Digression When doing Big-O analysis, we sometimes have to compute a series like: 1 + 2 + 3 + ... + (N-1) + N What is the complexity of this? Remember Gauss: 47/31
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Standard Analysis Techniques Sequence of Statements For a sequence of statements, compute their complexity functions individually and add them up for (j=0; j < N; j++) for (k =0; k < j; k++) sum = sum + j*k; O(N2) for (l=0; l < N; l++) sum = sum -l; O(N) printf("sum is now %f", sum); O(1) Total cost is O(N2) + O(N) +O(1) = O(N2) SUM RULE 48/31
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Standard Analysis Techniques Conditional StatementsWhat about conditional statements such as if (condition) statement1; else statement2;where statement1 runs in O(N) time and statement2 runs in O(N2) time?We use "worst case" complexity: among all inputs of size N, what is the maximumrunning time?The analysis for the example above is O(N2) 49/31
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Fast machine Vs Fast AlgorithmGet a 10 times fast computer, that can do a job in 103seconds for which the older machine took 104seconds .Comparing the performance of algorithms with timecomplexities T(n)s of n, n2 and 2n (technically not analgorithm) for different problems on both themachines.Question: Is it worth buying a 10 times fast machine? 50/31
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