Definition A transition system S is a pair of the form. where C is the set of configurations and T C C is a relation, the

Transcription

1 Transition system Definition A transition system S is a pair of the form = S T) (C, where C is the set of configurations and T C C is a relation, the transition Λ relation. Algorithms and Data Structures Chapter 1: Transition Systems 1

2 Sequences generated by a transition system Definition Let S = (C, T) be a transition system. S generates a set of sequences, S(S), defined as follows: 1. the finite sequence c 0, c 1,...,c n (for n 0) belongs to S(S) if (ι) c 0 C 2 (ιι) for all i with» 1» i n: (c i 1, c i 2 T ) 2. the infinite sequence c 0, c 1,...,c n,... belongs to S(S) if (ι) c 0 C 2 (ιι) for all i 1: (c i 1, c i 2 T ) Λ Algorithms and Data Structures Chapter 1: Transition Systems 2

3 Processes generated by a transition system Definition Let S = (C, T) be a transition system. The set of processes generated by S, written P (S), is the subset of S(S) containing 1. all infinite sequences of S(S) 2. all finite sequences c 0, c 1,...,c n (n 0) of S(S) for which it holds that there is no c 2 C with (c n, c) 2 T. The final configuration of a finite process is called a dead configuration. Λ Algorithms and Data Structures Chapter 1: Transition Systems 3

4 Football Football b] j»» 2 Ng 2 + b] Λ» b] + b]» Λ b] +» + b] b] Λ Λ + b] +» b] Λ b] b]» + Λ b] b] Λ» b] + + 1] Λ b]» b] + 1] b]» + Λ + Λ b] b] Transition system Configurations: f[t, X, a, 0 t 90, X fa, B, Rg, a, b [t, A, a, [t 2, B, a, if t 88 [t, A, a, [t 2, B, a 1, if t 88 [t, A, a, [t 1, B, a, if t 89 [t, A, a, [t 1, B, a 1, if t 89 [90, A, a, [90, R, a, [t, B, a, [t 2, A, a, if t 88 [t, B, a, [t 2, A, a, b if t 88 [t, B, a, [t 1, A, a, if t 89 [t, B, a, [t 1, A, a, b if t 89 [90, B, a, [90, R, a, Algorithms and Data Structures Chapter 1: Transition Systems 4

5 Induction principle Induction principle Let P(0), P(1),...,P(n),... bestatements. If P(0) a) is true b) for all n 0 it holds P(n) that implies + P(n 1), P(n) then is true for all n 0. Algorithms and Data Structures Chapter 1: Transition Systems 5

6 Invariance principle Invariance principle for transition systems Let S = (C, T) be a transition system and let c 0 2 C be a configuration. If I(c) is a statement about the configurations of the system, the following holds. If a) I(c 0 ) is true b) for all (c, c 0 ) 2 T it holds that I(c) implies I(c 0 ) then I(c) is true for any configuration c that occurs in a sequence starting with c 0. Algorithms and Data Structures Chapter 1: Transition Systems 6

7 Termination principle Termination principle for transition systems Let S = (C, T) be a transition system and let µ : C! N be a function. If for all (c, c 0 ) 2 T it holds that µ(c) > µ(c 0 ) then all processes in P (S) are finite. Algorithms and Data Structures Chapter 1: Transition Systems 7

8 Nim Nim Bg n] Λ 2] n] Λ 1] n] Λ 2] 1] n] Λ Transition system Configurations: fa, N [A, [B, n if n 2 [A, [B, n if n 1 [B, [A, n if n 2 [B, [A, n if n 1 Algorithms and Data Structures Chapter 1: Transition Systems 8

9 Towers of Hanoi Transition system Hanoi(n) Configurations: f[a, B, C] j fa, B, Cg a partition of f1,..., ngg [A,B,C] Λ [An frg,b[ frg,c] if (r = min A) ^ (r < min B) [A, B, C] Λ [A n frg, B, C [ frg] if (r = min A) ^ (r < min C) [A, B, C] Λ [A [ frg, B n frg, C] if (r = min B) ^ (r < min A) [A, B, C] Λ [A, B n frg, C [ frg] if (r = min B) ^ (r < min C) [A, B, C] Λ [A [ frg, B, C n frg] if (r = min C) ^ (r < min A) [A, B, C] Λ [A, B [ frg, C n frg] if (r = min C) ^ (r < min B) Algorithms and Data Structures Chapter 1: Transition Systems 9

10 Euclid s algorithm Transition Euclid system Configurations: n] j f[m, m, 1g n n] Λ [m, [m n] n, if > m n [m, n] Λ [m, n m] if m < n Algorithms and Data Structures Chapter 1: Transition Systems 10

11 Expressions Transition Expressions system Configurations: f0, 1, +, E, T, )g (, αeβ Λ αtβ Λ Λ αeβ αt+eβ Λ αtβ α0β Λ αtβ α1β Λ αtβ α(e)β Algorithms and Data Structures Chapter 1: Transition Systems 11

12 Expressions (context-free) Transition Expressions system Configurations: f0, 1, +, E, T, )g (, E Λ T, T+E Λ Λ T 0, (E) 1, Algorithms and Data Structures Chapter 1: Transition Systems 12

13 Graph coloring Transition GraphColoring system Configurations: Danish graphs Λ Λ if there are no pink nodes Λ Algorithms and Data Structures Chapter 1: Transition Systems 13

14 Red-black tree Definition A red-black tree is binary search tree in which all internal nodes are colored either red or black, in which the leaves are black, and Invariant I 2 Each red node has a black parent. Invariant I 3 There is the same number of black nodes on all paths from the root to a leaf Λ Algorithms and Data Structures Chapter 1: Transition Systems 14

15 Insertion Illegitimate red node: x Invariant I2 : Each legitimate red node has a black parent. 0 Algorithms and Data Structures Chapter 1: Transition Systems 15

16 Insertion: transitions 1 and 2 The illegitimate node is the root of the tree: x x 1 2 Λ 1 2 The illegitimate node has a black father: A A x Λ 3 x Algorithms and Data Structures Chapter 1: Transition Systems 16

17 Insertion: transitions 3.1 and 3.2 The illegitimate node has a red father and a red uncle: A A B C B C Λ x x A A B C B C Λ 1 x x Algorithms and Data Structures Chapter 1: Transition Systems 17

18 Insertion: transitions 4.1 and 4.2 The illegitimate node has a red father and a black uncle: x B A C Λ B x A C x B A C Λ x B A C Algorithms and Data Structures Chapter 1: Transition Systems 18

19 Deletion Illegitimate black node: Invariant I 0 1 The tree satisfies I 1 if we remove the illegitimate node. Invariant I 0 2 Each red node has a legitimate black father. Algorithms and Data Structures Chapter 1: Transition Systems 19

20 Deletion: transition 1 The illegitimate node is the root: Λ Algorithms and Data Structures Chapter 1: Transition Systems 20

21 Deletion: transitions 2 and 3 The illegitimate node has a red father and a red closer nephew: 1 A C 2 3 B 4 Λ A C B The illegitimate node has a red father and a black closer nephew: A B B 1 C 4 Λ A 4 1 C Algorithms and Data Structures Chapter 1: Transition Systems 21

22 Deletion: transitions 4.1 and 4.2 The illegitimate node has a black father, a black sibling and one red nephew: 1 1 A C 2 3 A 2 B B 4 C 3 4 Λ Λ A C B A B C Algorithms and Data Structures Chapter 1: Transition Systems 22

23 Deletion: transition 5 The illegitimate node has a black father, a black sibling and two black nephews 1 A C 2 3 B 4 D 5 Λ 1 A B C D 5 Algorithms and Data Structures Chapter 1: Transition Systems 23

24 Deletion: transition 6 The illegitimate node has a black father and a red sibling: B A A B Λ Algorithms and Data Structures Chapter 1: Transition Systems 24

25 The transition system Commands Transition system Commands Configurations: f[c, σ] j C a command-sequence, σ a stateg [x ψ e; C 0,σ] Λ [C 0,σhx:ei] b then C [if 1 else else C 2 ; C 0,σ] Λ [C 1 ; C 0,σ] σ(b) = true if σ(b) = false if b then C [if 1 else else C 2 ; C 0,σ] Λ [C 2 ; C 0,σ] b do [ do C; C 0,σ] Λ [C; b do do C; C 0,σ] if σ(b) = true b do [ do C; C 0,σ] Λ [C 0,σ] if σ(b) = false Algorithms and Data Structures Chapter 2: Algorithm Theory 25

26 Euclid s algorithm n) = ) 6= do Algorithm Euclid(m, Input : m, n 1 Output : r gcd(m 0, n 0 Method m n : if do m > n then if ψ m m n then else r ψ m n ψ n m; Algorithms and Data Structures Chapter 2: Algorithm Theory 26

27 Execution of Euclid(72,45) Step# Action State# m n r m<>n m > n m < m-n m<>n m > n n < n-m m<>n m > n m < m-n m<>n m > n n < n-m m<>n r < m

28 Correctness A( ) Algorithm Input : In Output : Out Method : C Definition The A algorithm is correct if any process for the transition Commands system starting in a configuration [C,σ], where σ satisfies In, is finite and ends with a configuration σ satisfying Out. 0 Λ Algorithms and Data Structures Chapter 2: Algorithm Theory 27

29 Decorations b do [fig do C; C 0,σ] Λ [C; fig b do do C; C 0,σ] if σ(b) = true. Definition An assertion U of a decorated algorithm is valid for a process if for all configurations of the [fugc,σ] form in the process, the assertion U is satisfied by the state Λ σ. Algorithms and Data Structures Chapter 2: Algorithm Theory 28

30 Euclid s algorithm (decorated) n) = ) 6= do Algorithm Euclid(m, Input : m, n 1 Output : r gcd(m 0, n 0 Method m n fig : if do m > n then if ψ m m n then else r ψ m n ψ n m; I : gcd(m, n) = gcd(m 0, n 0 ), Algorithms and Data Structures Chapter 2: Algorithm Theory 29

31 Validity A( ) Algorithm Input : In Output : Out Method : C Definition The A algorithm is valid if all its assertions are valid for all processes starting in a [C,σ] configuration where σ satisfies Λ In. Algorithms and Data Structures Chapter 2: Algorithm Theory 30

32 Proof-burdens fugcfvg For any state σ, ifσsatisfies U and the execution of C in σ leads to σ 0 (ie. [C,σ] Λ Λ σ 0 ), then σ 0 must satisfy V. Algorithms and Data Structures Chapter 2: Algorithm Theory 31

33 Proof principle for simple proof-burdens Proof principle for simple proof-burdens Let C = c 1 ; ;c k be a sequence of assignments and let x 1,...,x n be the variables of the algorithm. Suppose that C executed in σ leads to σ 0. Then the proofburden fugcfvg is proved by proving the implication U(x 1,...,x n ) ) V(x 0 1,...,x 0 n ) where x1 n are the values of the variables in σ 0 functions of their values in σ. 0,...,x0 expressed as Algorithms and Data Structures Chapter 2: Algorithm Theory 32

34 Proof principle for compound proof-burdens Proof principle for compound proof-burdens A proof-burden of the form fugc 1 ; C 2 fvg gives rise to the proof-burdens fugc 1 fwg and fwgc 2 fvg. A proof-burden of the fugif form the proof-burdens b then C if 1 else fu ^ bgc 1 fvg and fu ^ :bgc 2 fvg. A proof-burden of the fug form proof-burdens b do ) U I (basis) ^ bgcfig fi (invariance) ^ I ) :b V (conclusion). else C 2 fvg gives rise to do CfVg gives rise to the Algorithms and Data Structures Chapter 2: Algorithm Theory 33

35 Termination A( ) Algorithm Input : In Output : Out Method : C Definition We say A that terminates if any process starting in a configuration [C,σ], where σ satisfies In, is Λ finite. Algorithms and Data Structures Chapter 2: Algorithm Theory 34

36 Termination principle Termination principle for algorithms Let x 1,...,x n be the variables of an algoritm A. A terminates if for every loop in its method with I as valid invariant, there exists an integer-valued function µ(x 1,...,x n ) satisfying a) I ) µ(x 1,...,x n ) 0 fig b do C b) I ^ b ) µ(x 1,...,x n ) > µ(x 0 1,...,x0 n ) 0 where x 0 1,...,x0 n are the values of the variables after an iteration expressed as functions of their values before. Algorithms and Data Structures Chapter 2: Algorithm Theory 35

37 Extended version of Euclid s algorithm n) = )) ^ = )) ψ ψ Algorithm ExtendedEuclid(m, Input : m, n 1 Output : (r gcd(m 0, n 0 (s lcm(m 0, n 0 Method : p m; q n; m > n then if ψ m m n; ψ p + p q if then fig do m 6= n do n ψ n m; q ψ q + p; r ψ m; s ψ (p + q)=2 else I : (mq + np = 2m 0 n 0 ) ^ (gcd(m, n) = gcd(m 0, n 0 )). Algorithms and Data Structures Chapter 2: Algorithm Theory 36

38 Factorial Factorial(n) = ψ Algorithm Input : n 0 Output : r n 0! Method : r 1; n 6= 0 do ψ r Λ r n; ψ n n 1; fig do I : (r = n 0!=n!) ^ (n 0 n 0). Algorithms and Data Structures Chapter 2: Algorithm Theory 37

39 Power sum Algorithm n) PowerSum(x, Input : 6= 0) ^ (x 0) (n Constants: x, n Output : = ± r i=0 n xi Method : ψ r 1; ψ m 0; m 6= n do ψ r Λ r + x 1; ψ m + m 1; fig do I : (r = ± m i=0x i ) ^ (n m 0). Algorithms and Data Structures Chapter 2: Algorithm Theory 38

40 Finding maximum in an array Algorithm ArrayMax(A) true = ψ ψ Input : Constants: A Output : r max A Method : r 1; i 0; fig do i 6= jaj do if r < A[i] then r ψ A[i]; ψ i + i 1 I : (0» i» jaj) ^ (r = max A[0..i]) µ(a, i, r) = jaj i Algorithms and Data Structures Chapter 2: Algorithm Theory 39

41 Time complexity ) Algorithm A(x 1,...,x n Input : In Output : Out Method : C Definition The time complexity of A is the function TJAK taking a state σ satisfying In to the length of the process starting at [C,σ]. Λ We can regard TJAK as a function of the input parameters x 1,...,x n Algorithms and Data Structures Chapter 2: Algorithm Theory 40

42 Time complexity for ArrayMax Steps ψ r 1; ψ i 0; 2 i 6= jaj do jaj + 1 fig if r < A[i] then r ψ A[i]; between jaj and 2jAj ψ i + i jaj 1 Total number of steps: 3jAj + 3» TJArrayMaxK(A)» 4jAj + 3 More precisely, TJArrayMaxK(A) equals 3jAj + 3 plus the number of times A[i] is strictly larger than max A[0..i], with i running through the indices 0,..., jaj 1. Algorithms and Data Structures Chapter 2: Algorithm Theory 41

43 Time complexity for Euclid if then m ψ m n then if m > n then do m 6= n do else n ψ n m m n T m n T m n T m n T m T n Algorithms and Data Structures Chapter 2: Algorithm Theory 42

44 Worst-case time complexity ) Algorithm A(x 1,...,x n Input : In Output : Out Method : C Definition Let size be a function mapping states satisfying In to non-negative integers. The worst-case time complexity A of is the function mapping n 0tothe maximum TJAK(σ) of for states σ (satisfying In) size(σ) = with Λ n. We can regard size as a function of the input parameters x 1,...,x n, and write TJArrayMaxK(A) = 4jAj O(jAj) Algorithms and Data Structures Chapter 2: Algorithm Theory 43

45 Exponentiation in linear time Algorithm p) LinExp(x, Input : p 0 Constants: x, p Output : = r x p Method : ψ r 1; ψ q p; q > 0 do ψ r Λ r x; ψ q q 1 fig do I : (rx q = x p ) ^ (q 0) µ(x, p, r, q) = q Algorithms and Data Structures Chapter 3: Fundamental algorithms 44

46 Exponentiation in logarithmic time Algorithm p) LogExp(x, Input : p 0 Constants: x, p Output : = r x p Method : ψ r 1; ψ q p; ψ h x; q even then if ψ q q=2; ψ h Λ h h if then fig do q > 0 do else q ψ q 1; r ψ r Λ h I : (rh q = x p ) ^ (q 0) µ(x, p, r, q, h) = q Algorithms and Data Structures Chapter 3: Fundamental algorithms 45

47 Scanning A i I : (0» i» jaj) ^ I 0 µ(a, i,...) = jaj i fi init fl; i ψ 0; fig do i 6= jaj do update fl; i ψ i + 1; fi end fl fi Algorithms and Data Structures Chapter 3: Fundamental algorithms 46

48 Linear search Algorithm s) LinearSearch(A, Input true : Constants: A, s Output :» (0» jaj) ^ r 62 A[0..r]) ^ (s = jaj _ A[r] = s) (r Method : ψ r 0; (r 6= jaj) ^ (A[r] 6= s) do ψ r + r 1; fig do I : (0» r» jaj) ^ (s 62 A[0..r]) µ(a, s, r) = jaj r Algorithms and Data Structures Chapter 3: Fundamental algorithms 47

49 Binary search: specification and invariant Algorithm s) BinarySearch(A, Input : A sorted Constants: A, s Output :» (0» jaj) ^ (A[0..r] < s) ^ r» A[r..jAj]) (s A low high low ψ 0; high ψ jaj; fig do low 6= high do narrow the gap fl; fi ψ r low Algorithms and Data Structures Chapter 3: Fundamental algorithms 48

50 Binary search: the algorithm Algorithm s) BinarySearch(A, Input : A sorted Constants: A, s Output :» (0» jaj) ^ (A[0..r] < s) ^ r» A[r..jAj]) (s Method : ψ low 0; ψ high jaj; low 6= high do ψ m + (low high)=2; fig if A[m] < s then do then low ψ m + 1 else r ψ low high ψ m; I : (0» low» high» jaj) ^ (A[0..low] < s) ^ (s» A[high..jAj]) µ(a, s, low, high, m, r) = high low Algorithms and Data Structures Chapter 3: Fundamental algorithms 49

51 Insertion sort InsertionSort(A) Algorithm Input true : ) ^ sorted) ψ Output : (A perm A 0 (A Method : i 0; i 6= jaj do ψ j i; fig fjg do do ( j 6= 0) ^ (A[ j 1] > A[ j]) do j 1] $ A[ j]; A[ ψ j j 1; ψ i + i 1 I : (A perm A 0 ^ (0» i» jaj) ^ (A[0..i] sorted) ) J : (A perm A 0 ^ (0» j» i < jaj) ^ (A[0..j],A[ j..i + 1] sorted) ) µ I (A, j) = jaj i, i µ J (A, j) = i, j Algorithms and Data Structures Chapter 3: Fundamental algorithms 50

52 Merge sort Algorithm MergeSort(A) Input Output Method : true : (A perm A 0 ) ^ (A sorted) if : then B ψ A[0..jAj=2]; C ψ A[jAj=2..jAj]; MergeSort(B); MergeSort(C); fwgmerge(a, B, C) then if jaj > 1 then Algorithm Merge(A, C) B, Input (jaj = jbj + jcj) ^ : (B, sorted) C Constants: B, C Output : (A BC) ^ perm sorted) (A W : (A = A 0 ) ^ (B perm A 1 ) ^ (C perm A 2 ) ^ (B, C sorted) = jaj µ(a) Algorithms and Data Structures Chapter 3: Fundamental algorithms 51

53 Merge sort: time complexity, assuming n a power of 2 level 0 level 1 level 2 level 3 level 4 Time spent at level i (for 0» i» log n): 2 i n=2 i = n time units. Algorithms and Data Structures Chapter 3: Fundamental algorithms 52

54 Merge: specification Algorithm Merge(A, C) B, Input (jaj = jbj + jcj) ^ : (B, sorted) C Constants: B, C Output : (A BC) ^ perm sorted) (A So, with jaj = 11 and we should obtain B = [1, 1, 4, 7, 8] and C = [1, 2, 2, 3, 4, 7] A = [1, 1, 1, 2, 2, 3, 4, 4, 7, 7, 8]. Algorithms and Data Structures Chapter 3: Fundamental algorithms 53

55 Merge: scanning invariant A k=i+j B i C j I (jaj = jbj + jcj) ^ :» (0» jbj) ^ i» (0» jcj) ^ j + j] (A[0..i B[0..i]C[0..j]) ^ perm + j] sorted) (A[0..i fi init fl; fig do i + j 6= jaj do update fl; fi end fl fi Algorithms and Data Structures Chapter 3: Fundamental algorithms 54

56 Merge: the algorithm Algorithm Merge(A, C) B, Input (jaj = jbj + jcj) ^ : (B, sorted) C Constants: B, C Output : (A BC) ^ perm sorted) (A Method : ψ i 0; ψ j 0; (i < jbj) ^ ( j = jcj _ B[i]» C[ j]) then if + j] ψ A[i B[i]; ψ i + i 1 if then fig do i + j 6= jaj do else A[i + j] ψ C[ j]; j ψ j + 1 I (jaj = jbj + jcj) ^ :» (0» jbj) ^ i» (0» jcj) ^ j + j] (A[0..i B[0..i]C[0..j]) ^ perm + j] sorted) (A[0..i µ(a, B, C, i, j) = jaj (i + j) Algorithms and Data Structures Chapter 3: Fundamental algorithms 55

57 Dutch flag: specification A w b Algorithm s) DutchFlag(A, Input true : Constants: s Output : (A perm A 0 ^ (0» w» b» jaj) ^ ) (A[0..w] < s) ^ (A[w..b] = s) ^ (A[b..jAj] > s) Algorithms and Data Structures Chapter 3: Fundamental algorithms 56

58 I : (A perm A 0 ^ (0» w» b» i» jaj) ^ ) < s) ^ (A[w..b] = s) ^ (A[b..i] > s) (A[0..w] Dutch flag: scanning invariant A w b i fi init fl; i ψ 0; fig do i 6= jaj do update fl; i ψ i + 1; fi end fl fi Algorithms and Data Structures Chapter 3: Fundamental algorithms 57

59 Dutch flag: basis ftruegfi init fl; i ψ 0fIg A w=b=i fi init fl = w ψ 0; b ψ 0 Algorithms and Data Structures Chapter 3: Fundamental algorithms 58

60 Dutch flag: invariance fi ^ i 6= jajgfi update fl; i ψ i + 1fIg fi update fl = if A[i] < s then if fi red fl else if A[i] = s then fi white fl else fi blue fl 1: ^ fi 6= jaj) ^ (A[i] < s)gfi red (i fl; ψ i + 1fIg i 2: ^ fi 6= jaj) ^ (A[i] = s)gfi white (i fl; ψ i + 1fIg i 3: ^ fi 6= jaj) ^ (A[i] > s)gfi blue (i fl; ψ i + 1fIg i Algorithms and Data Structures Chapter 3: Fundamental algorithms 59

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62 Dutch flag: invariance, white fi ^ (i 6= jaj) ^ (A[i] = s)gfi white fl; i ψ i + 1fIg A w b i fi white fl = A[i] $ A[b]; b ψ b + 1 Algorithms and Data Structures Chapter 3: Fundamental algorithms 61

63 Dutch flag: invariance, blue fi ^ (i 6= jaj) ^ (A[i] > s)gfi blue fl; i ψ i + 1fIg A w b i fi blue fl = λ Algorithms and Data Structures Chapter 3: Fundamental algorithms 62

64 Dutch flag: conclusion fi ^ (i = jaj)gfi end flfoutg A w b i fi end fl = λ Algorithms and Data Structures Chapter 3: Fundamental algorithms 63

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66 Dutch flag: from low to high A low w b high Algorithm DutchFlag(A, low, s) high, Input :» 0» low» jaj high Constants: low, high, A[0..low], A[high..jAj], s Output (A[low..high] : perm A 0 ^ [low..high])» (low» w» high) ^ b (A[low..w] < s) ^ (A[w..b] = s) ^ (A[b..high] > s) Algorithms and Data Structures Chapter 3: Fundamental algorithms 65

67 Quick sort Algorithm QuickSort(A, high) low, Input :» 0» low» jaj high Constants: low, high, A[high..jAj] A[0..low], Output (A[low..high] : perm A 0 ^ (A[low..high] sorted) [low..high]) Method if : > then 1 ψ r random(low, high); b) ψ (w, DutchFlag(A, low, high, A[r]); QuickSort(A, low, w); QuickSort(A, high) b, µ(a, low, high) = high low Algorithms and Data Structures Chapter 3: Fundamental algorithms 66

68 Quick select Algorithm QuickSelect(A, low, k) high, Input :» (0» low < k» jaj) high Constants: low, high, A[0..low], A[high..jAj], k Output (A[low..high] : perm A 0 ^ [low..high]) Method» A[k]» A[k..high]) (A[low..k] : ψ r random(low, high); b) ψ (w, DutchFlag(A, low, high, A[r]); then QuickSelect(A, low, w, k) if then if k < w then then QuickSelect(A, b, high, k) else if then if k b then µ(a, low, high) = high low Algorithms and Data Structures Chapter 3: Fundamental algorithms 67

69 Maximal subsum MaxSubsum(A) Algorithm Input true : = ms(a) fi init ψ Constants: A Output : r Method : fl; i 0; fig do i 6= jaj do update fl; i ψ i + 1; fi end fl fi I : (0» i» jaj) ^ (r = ms(a[0..i])) ^ Algorithms and Data Structures Chapter 3: Fundamental algorithms 68

70 Maximal subsum: maintaining the invariant ms(a[0..i + 1]) is the maximum of the maximal subsum obtained by not using A[i], and the maximal subsum obtained by using A[i] So, we remember also the maximal right subsum mrs(a[0..i]) I : (0» i» jaj) ^ (r = ms(a[0..i])) ^ (h = mrs(a[0..i])) Algorithms and Data Structures Chapter 3: Fundamental algorithms 69

71 Maximal subsum: the algorithm MaxSubsum(A) Algorithm Input true : = ms(a) ψ ψ ψ Constants: A Output : m Method : r 0; h 0; i 0; i 6= jaj do ψ h + maxfh A[i],0g; ψ r maxfr, hg; ψ i + i 1 fig do µ(a, r, h, i) = jaj i Algorithms and Data Structures Chapter 3: Fundamental algorithms 70

72 Longest monotone sequence LLMS(A) Algorithm Input true : Constants: A Output : = llms(a) r Method fi init : fl; ψ i 0; fig do i 6= jaj do update fl; i ψ i + 1; fi end fl fi I : (0» i» jaj) ^ (r = llms(a[0..i])) ^ Algorithms and Data Structures Chapter 3: Fundamental algorithms 71

73 I :» (0» jaj) ^ i = llms(a[0..i])) ^ (r = ms (B[l] l for 0» l» jaj) (A[0..i]) Longest monotone sequence: maintaining the invariant In the i th iteration of the loop, B[l] for 1» l» jaj must contain the necessary information about a monotone sequence from A[0..i] of length l whose last element is minimal (if there is no sequence of length l,wejust record that). ms l (A[0..i]) = 8 >< >: if = 1 l 0 minimal last element in a monotone sequence of if such exists A[0..i] length l from 1 otherwise Algorithms and Data Structures Chapter 3: Fundamental algorithms 72

74 Longest monotone sequence: the algorithm LLMS(A) Algorithm Input true : Constants: A Output : = llms(a) r Method : ψ [jaj + B B[0] ψ 1:1]; 1; ψ r 0; ψ i 0; i 6= jaj do ψ BinarySearch l (B, A[i]); 0 fig do B[l] ψ A[i]; r ψ maxfr, lg; i ψ i + 1 µ(a, B, r, i) = jaj i BinarySearch Algorithm (B, 0 s) Input : B sorted Constants: B, s Output :» (0» jbj) ^ (B[0..l]» s) ^ l < B[l..jBj]) (s Algorithms and Data Structures Chapter 3: Fundamental algorithms 73