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This recap discusses the concepts of Oh, Omega, and Theta notation in asymptotic analysis, their properties, and how they are used to compare functions. It also introduces the divide and conquer approach and provides examples of its application in solving recurrence relations in algorithms like merge sort and the max subarray problem.
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Recap: Oh, Omega, Theta • Oh (like ≤) • Omega (like ≥) • Theta(like =) O(n) is asymptotic upper bound 0 ≤ f(n) ≤ cg(n) Ω(n) is asymptotic lower bound 0 ≤ cg(n) ≤ f(n) Θ(n) is asymptotic tight bound 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n) CSC317
More on Oh, Omega, Theta Theorem: f(n) = Θ(n) if and only if (iff) f(n) = O(n)and f(n) = Ω(n) Question: Is f(n) = n2 + 5 Ω(n3) ? Answer: NO (why?)! CSC317
Answer: NO! Proof by contradiction Question: Is f(n) = n2 + 5 Ω(n3) ? Definition:Ω (g(n)) = f(n): There exist positive constants c, such that 0 ≤ cg(n) ≤ f(n) for all n ≥ n0 Therefore: If f(n) = Ω(n3), then there exists n0; c such that for all n ≥ n0 n2 + 5 ≥ cn3 Remember from before: n2 + 5n2 ≥ n2+ 5 n2 + 5n2 ≥ cn3 6 ≥ cn Can’t be true for all n ≥ n0 CSC317
Some properties of Oh, Omega, Theta Transitivity: f(n) = Θ(g(n)) and g(n) = Θ(h(n)) then f(n) = Θ(h(n)) (same for O and Ω?) Reflexivity: f(n) = Θ(f(n)) (same for O and Ω?) f(n) = Θ(g(n)) iff g(n) = Θ(f(n)) Symmetry: (same for O and Ω?) CSC317
Final thoughts: Oh, Omega, Theta • Useful comparison formula of functions in book (section3.2) CSC317
What kind of recurrences arise in algorithms and how do we solve more generally (than what we saw for merge sort)? • More recurrence examples • Run time not always intuitive, so need tools Divide and Conquer approach Max Subarray Problem: Can buy stock once, sell stock once. Want to maximize profit; allowed to look into the future CSC317
Max Subarray Problem: Can buy stock once, sell stock once. Want to maximize profit; allowed to look into the future Sell Buy Buy low, sell high … CSC317
Brute force: Try every possible pair of buy and sell dates Hmm, can we do better? CSC317
Let’s reframe the problem as greatest sum of any contiguous array Efficiency? Still brute force Divide and conquer anybody? CSC317
Where could max subarray be? low low high high middle middle i j i j i j CSC317
Where could max subarray be? low high middle DIVIDE chop subarray in two equal subarrays A[low,…,middle] and A[middle+1,..., high] 2. CONQUERfind max of subarrays A[low,…,middle] and A[middle+1,..., high] 3. COMBINE find the best solution of: a. the two solutions found in conquer step b. solution of subarray crossing the midpoint CSC317
Keep recursing until low=high (one element left) DIVIDE chop subarray in two equal subarrays A[low,…,middle] and A[middle+1,..., high] 2. CONQUER find max of subarrays A[low,…,middle] and A[middle+1,..., high] 3. COMBINE find the best solution of: a. the two solutions found in conquer step b. solution of subarray crossing the midpoint low high middle i j CSC317
i j low high middle • Start from the middle • Go to the left until hit max sum. • Go to the right until hit max sum. • Return total left and right sum. • COMPLEXITY? Θ(n) CSC317
float recmax(int l, int u) if (l > u) /* zero elements */ return 0; if (l == u) /* one element */ return max(0, A[l]); m = (l+u) / 2; /* find max crossing to left */ lmax = sum = 0; for (i = m; i ≥ l; i--) { sum += A[i]; if (sum > lmax) lmax = sum; } /* find maxcrossingtoright */ rmax = sum = 0; for (i = m+1; i ≤ u; i++) { sum += A[i]; if (sum > rmax) rmax = sum; } returnmax(max(recmax(l, m), recmax(m+1, u)), lmax + rmax); CSC317
COSTS: DIVIDE CONQUER COMBINE T = Θ(1) T = 2T(n/2) T = Θ(n)+Θ(1) comparison Subarray crossing TOTAL : T(n) = 2T(n/2) + Θ(n) = Θ(n log n) Merge sort anyone? CSC317
CLASSICAL EXAMPLE: MATRIX MULTIPLICATION Run time: O(n3) in the naïve implementation 1. n = A.rows 2. Let C be a new n by n matrix 3.for i=1 to n 4.forj=1 to n 5. cij = 0 6. fork=1 to n 7. cij= cij+ aikbkj 8. returnC Can we do better? CSC317
