Introduction to Algorithms 6.046J/18.401J/SMA5503 Lecture 5 Prof. Erik Demaine Introduction to Algorithms Day 8 L5.2? 2001 by Charles E. Leiserson How fast can we sort? All the sorting algorithms we have seen so far are comparison sorts: only use comparisons to determine the relative order of elements. ? E.g., insertion sort, merge sort, quicksort, heapsort. The best worst-case running time that we’ve seen for comparison sorting is O(n lg n) . Is O(n lg n) the best we can do? Decision trees can help us answer this question. Introduction to Algorithms Day 8 L5.3? 2001 by Charles E. Leiserson Decision-tree example 1:2 1:2 2:3 2:3 123 123 1:3 1:3 132 132 312 312 1:3 1:3 213 213 2:3 2:3 231 231 321 321 Each internal node is labeled i:j for i, j ∈ {1, 2,…, n}. ?The left subtree shows subsequent comparisons if a i ≤ a j . ?The right subtree shows subsequent comparisons if a i ≥ a j . Sort ?a 1 , a 2 , …, a n ? Introduction to Algorithms Day 8 L5.4? 2001 by Charles E. Leiserson Decision-tree example 1:2 1:2 2:3 2:3 123 123 1:3 1:3 132 132 312 312 1:3 1:3 213 213 2:3 2:3 231 231 321 321 Each internal node is labeled i:j for i, j ∈ {1, 2,…, n}. ?The left subtree shows subsequent comparisons if a i ≤ a j . ?The right subtree shows subsequent comparisons if a i ≥ a j . 9 ≥ 4 Sort ?a 1 , a 2 , a 3 ? = ? 9, 4, 6 ?: Introduction to Algorithms Day 8 L5.5? 2001 by Charles E. Leiserson Decision-tree example 1:2 1:2 2:3 2:3 123 123 1:3 1:3 132 132 312 312 1:3 1:3 213 213 2:3 2:3 231 231 321 321 Each internal node is labeled i:j for i, j ∈ {1, 2,…, n}. ?The left subtree shows subsequent comparisons if a i ≤ a j . ?The right subtree shows subsequent comparisons if a i ≥ a j . 9 ≥ 6 Sort ?a 1 , a 2 , a 3 ? = ? 9, 4, 6 ?: Introduction to Algorithms Day 8 L5.6? 2001 by Charles E. Leiserson Decision-tree example 1:2 1:2 2:3 2:3 123 123 1:3 1:3 132 132 312 312 1:3 1:3 213 213 2:3 2:3 231 231 321 321 Each internal node is labeled i:j for i, j ∈ {1, 2,…, n}. ?The left subtree shows subsequent comparisons if a i ≤ a j . ?The right subtree shows subsequent comparisons if a i ≥ a j . 4 ≤ 6 Sort ?a 1 , a 2 , a 3 ? = ? 9, 4, 6 ?: Introduction to Algorithms Day 8 L5.7? 2001 by Charles E. Leiserson Decision-tree example 1:2 1:2 2:3 2:3 123 123 1:3 1:3 132 132 312 312 1:3 1:3 213 213 2:3 2:3 231 231 321 321 Each leaf contains a permutation ?π(1), π(2),…, π(n)? to indicate that the ordering a π(1) ≤ a π(2) ≤L≤ a π(n) has been established. 4 ≤ 6 ≤ 9 Sort ?a 1 , a 2 , a 3 ? = ? 9, 4, 6 ?: Introduction to Algorithms Day 8 L5.8? 2001 by Charles E. Leiserson Decision-tree model A decision tree can model the execution of any comparison sort: ? One tree for each input size n. ? View the algorithm as splitting whenever it compares two elements. ? The tree contains the comparisons along all possible instruction traces. ? The running time of the algorithm = the length of the path taken. ? Worst-case running time = height of tree. Introduction to Algorithms Day 8 L5.9? 2001 by Charles E. Leiserson Lower bound for decision- tree sorting Theorem. Any decision tree that can sort n elements must have height ?(n lg n) . Proof. The tree must contain ≥ n! leaves, since there are n! possible permutations. A height-h binary tree has ≤ 2 h leaves. Thus, n! ≤ 2 h . ∴ h ≥ lg(n!) (lg is mono. increasing) ≥ lg ((n/e) n ) (Stirling’s formula) = n lg n – n lg e = ?(n lg n) . Introduction to Algorithms Day 8 L5.10? 2001 by Charles E. Leiserson Lower bound for comparison sorting Corollary. Heapsort and merge sort are asymptotically optimal comparison sorting algorithms. Introduction to Algorithms Day 8 L5.11? 2001 by Charles E. Leiserson Sorting in linear time Counting sort: No comparisons between elements. ? Input: A[1 . . n], where A[ j]∈{1, 2, …, k} . ? Output: B[1 . . n], sorted. ? Auxiliary storage: C[1 . . k] . Introduction to Algorithms Day 8 L5.12? 2001 by Charles E. Leiserson Counting sort for i ← 1 to k do C[i] ← 0 for j ← 1 to n do C[A[ j]] ← C[A[ j]] + 1 ? C[i] = |{key = i}| for i ← 2 to k do C[i] ← C[i] + C[i–1] ? C[i] = |{key ≤ i}| for j ← n downto 1 do B[C[A[ j]]] ← A[ j] C[A[ j]] ← C[A[ j]] – 1 Introduction to Algorithms Day 8 L5.13? 2001 by Charles E. Leiserson Counting-sort example A: 4 4 1 1 3 3 4 4 3 3 B: 12345 C: 1234 Introduction to Algorithms Day 8 L5.14? 2001 by Charles E. Leiserson Loop 1 A: 4 4 1 1 3 3 4 4 3 3 B: 12345 C: 0 0 0 0 0 0 0 0 1234 for i ← 1 to k do C[i] ← 0 Introduction to Algorithms Day 8 L5.15? 2001 by Charles E. Leiserson Loop 2 A: 4 4 1 1 3 3 4 4 3 3 B: 12345 C: 0 0 0 0 0 0 1 1 1234 for j ← 1 to n do C[A[ j]] ← C[A[ j]] + 1 ? C[i] = |{key = i}| Introduction to Algorithms Day 8 L5.16? 2001 by Charles E. Leiserson Loop 2 A: 4 4 1 1 3 3 4 4 3 3 B: 12345 C: 1 1 0 0 0 0 1 1 1234 for j ← 1 to n do C[A[ j]] ← C[A[ j]] + 1 ? C[i] = |{key = i}| Introduction to Algorithms Day 8 L5.17? 2001 by Charles E. Leiserson Loop 2 A: 4 4 1 1 3 3 4 4 3 3 B: 12345 C: 1 1 0 0 1 1 1 1 1234 for j ← 1 to n do C[A[ j]] ← C[A[ j]] + 1 ? C[i] = |{key = i}| Introduction to Algorithms Day 8 L5.18? 2001 by Charles E. Leiserson Loop 2 A: 4 4 1 1 3 3 4 4 3 3 B: 12345 C: 1 1 0 0 1 1 2 2 1234 for j ← 1 to n do C[A[ j]] ← C[A[ j]] + 1 ? C[i] = |{key = i}| Introduction to Algorithms Day 8 L5.19? 2001 by Charles E. Leiserson Loop 2 A: 4 4 1 1 3 3 4 4 3 3 B: 12345 C: 1 1 0 0 2 2 2 2 1234 for j ← 1 to n do C[A[ j]] ← C[A[ j]] + 1 ? C[i] = |{key = i}| Introduction to Algorithms Day 8 L5.20? 2001 by Charles E. Leiserson Loop 3 A: 4 4 1 1 3 3 4 4 3 3 B: 12345 C: 1 1 0 0 2 2 2 2 1234 C': 1 1 1 1 2 2 2 2 for i ← 2 to k do C[i] ← C[i] + C[i–1] ? C[i] = |{key ≤ i}| Introduction to Algorithms Day 8 L5.21? 2001 by Charles E. Leiserson Loop 3 A: 4 4 1 1 3 3 4 4 3 3 B: 12345 C: 1 1 0 0 2 2 2 2 1234 C': 1 1 1 1 3 3 2 2 for i ← 2 to k do C[i] ← C[i] + C[i–1] ? C[i] = |{key ≤ i}| Introduction to Algorithms Day 8 L5.22? 2001 by Charles E. Leiserson Loop 3 A: 4 4 1 1 3 3 4 4 3 3 B: 12345 C: 1 1 0 0 2 2 2 2 1234 C': 1 1 1 1 3 3 5 5 for i ← 2 to k do C[i] ← C[i] + C[i–1] ? C[i] = |{key ≤ i}| Introduction to Algorithms Day 8 L5.23? 2001 by Charles E. Leiserson Loop 4 A: 4 4 1 1 3 3 4 4 3 3 B: 3 3 12345 C: 1 1 1 1 3 3 5 5 1234 C': 1 1 1 1 2 2 5 5 for j ← n downto 1 do B[C[A[ j]]] ← A[ j] C[A[ j]] ← C[A[ j]] – 1 Introduction to Algorithms Day 8 L5.24? 2001 by Charles E. Leiserson Loop 4 A: 4 4 1 1 3 3 4 4 3 3 B: 3 3 4 4 12345 C: 1 1 1 1 2 2 5 5 1234 C': 1 1 1 1 2 2 4 4 for j ← n downto 1 do B[C[A[ j]]] ← A[ j] C[A[ j]] ← C[A[ j]] – 1 Introduction to Algorithms Day 8 L5.25? 2001 by Charles E. Leiserson Loop 4 A: 4 4 1 1 3 3 4 4 3 3 B: 3 3 3 3 4 4 12345 C: 1 1 1 1 2 2 4 4 1234 C': 1 1 1 1 1 1 4 4 for j ← n downto 1 do B[C[A[ j]]] ← A[ j] C[A[ j]] ← C[A[ j]] – 1 Introduction to Algorithms Day 8 L5.26? 2001 by Charles E. Leiserson Loop 4 A: 4 4 1 1 3 3 4 4 3 3 B: 1 1 3 3 3 3 4 4 12345 C: 1 1 1 1 1 1 4 4 1234 C': 0 0 1 1 1 1 4 4 for j ← n downto 1 do B[C[A[ j]]] ← A[ j] C[A[ j]] ← C[A[ j]] – 1 Introduction to Algorithms Day 8 L5.27? 2001 by Charles E. Leiserson Loop 4 A: 4 4 1 1 3 3 4 4 3 3 B: 1 1 3 3 3 3 4 4 4 4 12345 C: 0 0 1 1 1 1 4 4 1234 C': 0 0 1 1 1 1 3 3 for j ← n downto 1 do B[C[A[ j]]] ← A[ j] C[A[ j]] ← C[A[ j]] – 1 Introduction to Algorithms Day 8 L5.28? 2001 by Charles E. Leiserson Analysis for i ← 1 to k do C[i] ← 0 Θ(n) Θ(k) Θ(n) Θ(k) for j ← 1 to n do C[A[ j]] ← C[A[ j]] + 1 for i ← 2 to k do C[i] ← C[i] + C[i–1] for j ← n downto 1 do B[C[A[ j]]] ← A[ j] C[A[ j]] ← C[A[ j]] – 1 Θ(n + k) Introduction to Algorithms Day 8 L5.29? 2001 by Charles E. Leiserson Running time If k = O(n), then counting sort takes Θ(n) time. ? But, sorting takes ?(n lg n) time! ? Where’s the fallacy? Answer: ? Comparison sorting takes ?(n lg n) time. ? Counting sort is not a comparison sort. ? In fact, not a single comparison between elements occurs! Introduction to Algorithms Day 8 L5.30? 2001 by Charles E. Leiserson Stable sorting Counting sort is a stable sort: it preserves the input order among equal elements. A: 4 4 1 1 3 3 4 4 3 3 B: 1 1 3 3 3 3 4 4 4 4 Exercise: What other sorts have this property? Introduction to Algorithms Day 8 L5.31? 2001 by Charles E. Leiserson Radix sort ? Origin: Herman Hollerith’s card-sorting machine for the 1890 U.S. Census. (See Appendix .) ? Digit-by-digit sort. ? Hollerith’s original (bad) idea: sort on most-significant digit first. ? Good idea: Sort on least-significant digit first with auxiliary stable sort. Introduction to Algorithms Day 8 L5.32? 2001 by Charles E. Leiserson Operation of radix sort 3 2 9 4 5 7 6 5 7 8 3 9 4 3 6 7 2 0 3 5 5 7 2 0 3 5 5 4 3 6 4 5 7 6 5 7 3 2 9 8 3 9 7 2 0 3 2 9 4 3 6 8 3 9 3 5 5 4 5 7 6 5 7 3 2 9 3 5 5 4 3 6 4 5 7 6 5 7 7 2 0 8 3 9 Introduction to Algorithms Day 8 L5.33? 2001 by Charles E. Leiserson ? Sort on digit t Correctness of radix sort Induction on digit position ? Assume that the numbers are sorted by their low-order t –1digits. 7 2 0 3 2 9 4 3 6 8 3 9 3 5 5 4 5 7 6 5 7 3 2 9 3 5 5 4 3 6 4 5 7 6 5 7 7 2 0 8 3 9 Introduction to Algorithms Day 8 L5.34? 2001 by Charles E. Leiserson ? Sort on digit t Correctness of radix sort Induction on digit position ? Assume that the numbers are sorted by their low-order t –1digits. 7 2 0 3 2 9 4 3 6 8 3 9 3 5 5 4 5 7 6 5 7 3 2 9 3 5 5 4 3 6 4 5 7 6 5 7 7 2 0 8 3 9 ? Two numbers that differ in digit t are correctly sorted. Introduction to Algorithms Day 8 L5.35? 2001 by Charles E. Leiserson ? Sort on digit t Correctness of radix sort Induction on digit position ? Assume that the numbers are sorted by their low-order t –1digits. 7 2 0 3 2 9 4 3 6 8 3 9 3 5 5 4 5 7 6 5 7 3 2 9 3 5 5 4 3 6 4 5 7 6 5 7 7 2 0 8 3 9 ? Two numbers that differ in digit t are correctly sorted. ? Two numbers equal in digit t are put in the same order as the input ? correct order. Introduction to Algorithms Day 8 L5.36? 2001 by Charles E. Leiserson Analysis of radix sort ? Assume counting sort is the auxiliary stable sort. ? Sort n computer words of b bits each. ? Each word can be viewed as having b/r base-2 r digits. Example: 32-bit word 8888 r = 8 ? b/r =4passes of counting sort on base-2 8 digits; or r = 16 ? b/r =2passes of counting sort on base-2 16 digits. How many passes should we make? Introduction to Algorithms Day 8 L5.37? 2001 by Charles E. Leiserson Analysis (continued) Recall: Counting sort takes Θ(n + k) time to sort n numbers in the range from 0 to k –1. If each b-bit word is broken into r-bit pieces, each pass of counting sort takes Θ(n + 2 r ) time. Since there are b/r passes, we have ( ) ? ? ? ? ? ? +Θ= r n r b bnT 2),( . Choose r to minimize T(n, b): ? Increasing r means fewer passes, but as r > lg n, the time grows exponentially.> Introduction to Algorithms Day 8 L5.38? 2001 by Charles E. Leiserson Choosing r ( ) ? ? ? ? ? ? +Θ= r n r b bnT 2),( Minimize T(n, b) by differentiating and setting to 0. Or, just observe that we don’t want 2 r > n, and there’s no harm asymptotically in choosing r as large as possible subject to this constraint. > Choosing r = lg n implies T(n, b) = Θ(bn/lg n) . ? For numbers in the range from 0 to n d –1, we have b = d lg n ? radix sort runs in Θ(dn) time. Introduction to Algorithms Day 8 L5.39? 2001 by Charles E. Leiserson Conclusions Example (32-bit numbers): ? At most 3 passes when sorting ≥ 2000 numbers. ? Merge sort and quicksort do at least ?lg 2000? = 11 passes. In practice, radix sort is fast for large inputs, as well as simple to code and maintain. Downside: Unlike quicksort, radix sort displays little locality of reference, and thus a well-tuned quicksort fares better on modern processors, which feature steep memory hierarchies. Introduction to Algorithms Day 8 L5.40? 2001 by Charles E. Leiserson Appendix: Punched-card technology ? Herman Hollerith (1860-1929) ? Punched cards ? Hollerith’s tabulating system ? Operation of the sorter ? Origin of radix sort ? “Modern” IBM card ? Web resources on punched- card technology Introduction to Algorithms Day 8 L5.41? 2001 by Charles E. Leiserson Herman Hollerith (1860-1929) ? The 1880 U.S. Census took almost 10 years to process. ? While a lecturer at MIT, Hollerith prototyped punched-card technology. ? His machines, including a “card sorter,” allowed the 1890 census total to be reported in 6 weeks. ? He founded the Tabulating Machine Company in 1911, which merged with other companies in 1924 to form International Business Machines. Introduction to Algorithms Day 8 L5.42? 2001 by Charles E. Leiserson Punched cards ? Punched card = data record. ? Hole = value. ? Algorithm = machine + human operator. Replica of punch card from the 1900 U.S. census: [Howells 2000] Introduction to Algorithms Day 8 L5.43? 2001 by Charles E. Leiserson Hollerith’s tabulating system ?Pantograph card punch ?Hand-press reader ?Dial counters ?Sorting box See figure from [Howells 2000]. Introduction to Algorithms Day 8 L5.44? 2001 by Charles E. Leiserson Operation of the sorter ? An operator inserts a card into the press. ? Pins on the press reach through the punched holes to make electrical contact with mercury- filled cups beneath the card. ? Whenever a particular digit value is punched, the lid of the corresponding sorting bin lifts. ? The operator deposits the card into the bin and closes the lid. ? When all cards have been processed, the front panel is opened, and the cards are collected in order, yielding one pass of a stable sort. Introduction to Algorithms Day 8 L5.45? 2001 by Charles E. Leiserson Origin of radix sort Hollerith’s original 1889 patent alludes to a most- significant-digit-first radix sort: “The most complicated combinations can readily be counted with comparatively few counters or relays by first assorting the cards according to the first items entering into the combinations, then reassorting each group according to the second item entering into the combination, and so on, and finally counting on a few counters the last item of the combination for each group of cards.” Least-significant-digit-first radix sort seems to be a folk invention originated by machine operators. Introduction to Algorithms Day 8 L5.46? 2001 by Charles E. Leiserson “Modern” IBM card So, that’s why text windows have 80 columns! See examples on the WWW Virtual Punch-Card Server. . ? One character per column. Introduction to Algorithms Day 8 L5.47? 2001 by Charles E. Leiserson Web resources on punched- card technology ? Doug Jones’s punched card index ? Biography of Herman Hollerith ? The 1890 U.S. Census ? Early history of IBM ? Pictures of Hollerith’s inventions ? Hollerith’s patent application (borrowed from Gordon Bell’s CyberMuseum) ? Impact of punched cards on U.S. history