Divide and Conquer Algorithms
|
|
|
- Ross Townsend
- 7 years ago
- Views:
Transcription
1 Divide and Conquer Algorithms T. M. Murali October 5 and 7, 2009
2 Divide and Conquer Algorithms Study three divide and conquer algorithms: Counting inversions. Finding the closest pair of points. Integer multiplication. First two problems use clever conquer strategies. Third problem uses a clever divide strategy.
3 Motivation Collaborative filtering: match one user s preferences to those of other users. Meta-search engines: merge results of multiple search engines to into a better search result.
4 Motivation Collaborative filtering: match one user s preferences to those of other users. Meta-search engines: merge results of multiple search engines to into a better search result. Fundamental question: how do we compare a pair of rankings?
5 Motivation Collaborative filtering: match one user s preferences to those of other users. Meta-search engines: merge results of multiple search engines to into a better search result. Fundamental question: how do we compare a pair of rankings? Suggestion: two rankings are very similar if they have few inversions.
6 Motivation Collaborative filtering: match one user s preferences to those of other users. Meta-search engines: merge results of multiple search engines to into a better search result. Fundamental question: how do we compare a pair of rankings? Suggestion: two rankings are very similar if they have few inversions. Assume one ranking is the ordered list of integers from 1 to n. The other ranking is a permutation a1, a 2,..., a n of the integers from 1 to n. The second ranking has an inversion if there exist i, j such that i < j but a i > a j. The number of inversions s is a measure of the difference between the rankings. Question also arises in statistics: Kendall s rank correlation of two lists of numbers is 1 2s/ (n(n 1)).
7 Counting Inversions Count Inversions INSTANCE: A list L = x 1, x 2,..., x n of distinct integers between 1 and n. SOLUTION: The number of pairs (i, j), 1 i < j n such x i > x j.
8 Counting Inversions Count Inversions INSTANCE: A list L = x 1, x 2,..., x n of distinct integers between 1 and n. SOLUTION: The number of pairs (i, j), 1 i < j n such x i > x j.
9 Counting Inversions: Algorithm How many inversions can be there in a list of n numbers?
10 Counting Inversions: Algorithm How many inversions can be there in a list of n numbers? Ω(n 2 ). We cannot afford to compute each inversion explicitly.
11 Counting Inversions: Algorithm How many inversions can be there in a list of n numbers? Ω(n 2 ). We cannot afford to compute each inversion explicitly. Sorting removes all inversions in O(n log n) time. Can we modify the Mergesort algorithm to count inversions?
12 Counting Inversions: Algorithm How many inversions can be there in a list of n numbers? Ω(n 2 ). We cannot afford to compute each inversion explicitly. Sorting removes all inversions in O(n log n) time. Can we modify the Mergesort algorithm to count inversions? Candidate algorithm: 1. Partition L into two lists A and B of size n/2 each. 2. Recursively count the number of inversions in A. 3. Recursively count the number of inversions in B. 4. Count the number of inversions involving one element in A and one element in B.
13 Counting Inversions: Conquer Step Given lists A = a 1, a 2,..., a m and B = b 1, b 2,... b m, compute the number of pairs a i and b j such a i > b j.
14 Counting Inversions: Conquer Step Given lists A = a 1, a 2,..., a m and B = b 1, b 2,... b m, compute the number of pairs a i and b j such a i > b j. Key idea: problem is much easier if A and B are sorted!
15 Counting Inversions: Conquer Step Given lists A = a 1, a 2,..., a m and B = b 1, b 2,... b m, compute the number of pairs a i and b j such a i > b j. Key idea: problem is much easier if A and B are sorted! Merge procedure: 1. Maintain a current pointer for each list. 3. Initialise each pointer to the front of the list. 4. While both lists are nonempty: 4.1 Let a i and b j be the elements pointed to by the current pointers. 4.2 Append the smaller of the two to the output list. 4.4 Advance the current pointer in the list that the smaller element belonged to. 5. Append the rest of the non-empty list to the output. 6. Return the merged list.
16 Counting Inversions: Conquer Step Given lists A = a 1, a 2,..., a m and B = b 1, b 2,... b m, compute the number of pairs a i and b j such a i > b j. Key idea: problem is much easier if A and B are sorted! Merge-and-Count procedure: 1. Maintain a current pointer for each list. 2. Maintain a variable count initialised to Initialise each pointer to the front of the list. 4. While both lists are nonempty: 4.1 Let a i and b j be the elements pointed to by the current pointers. 4.2 Append the smaller of the two to the output list. 4.3 If b j is the smaller, increment count by the number of elements remaining in A. 4.4 Advance the current pointer in the list that the smaller element belonged to. 5. Append the rest of the non-empty list to the output. 6. Return count and the merged list.
17 Counting Inversions: Conquer Step Given lists A = a 1, a 2,..., a m and B = b 1, b 2,... b m, compute the number of pairs a i and b j such a i > b j. Key idea: problem is much easier if A and B are sorted! Merge-and-Count procedure: 1. Maintain a current pointer for each list. 2. Maintain a variable count initialised to Initialise each pointer to the front of the list. 4. While both lists are nonempty: 4.1 Let a i and b j be the elements pointed to by the current pointers. 4.2 Append the smaller of the two to the output list. 4.3 If b j is the smaller, increment count by the number of elements remaining in A. 4.4 Advance the current pointer in the list that the smaller element belonged to. 5. Append the rest of the non-empty list to the output. 6. Return count and the merged list. Running time of this algorithm is O(m).
18 Counting Inversions: Final Algorithm
19 Counting Inversions: Final Algorithm Running time T (n) of the algorithm is O(n log n) because T (n) 2T (n/2) + O(n).
20 Counting Inversions: Correctness of Sort-and-Count Prove by induction. Strategy: every inversion in the data is counted exactly once.
21 Counting Inversions: Correctness of Sort-and-Count Prove by induction. Strategy: every inversion in the data is counted exactly once. Base case: n = 1. Inductive hypothesis: Algorithm counts number of inversions correctly for all sets of n 1 or fewer numbers. Inductive step: Pick an arbitrary k and l such that k < l but x k > x l. When is this inversion counted? k, l n/2 : k, l n/2 : k n/2, l n/2 :
22 Counting Inversions: Correctness of Sort-and-Count Prove by induction. Strategy: every inversion in the data is counted exactly once. Base case: n = 1. Inductive hypothesis: Algorithm counts number of inversions correctly for all sets of n 1 or fewer numbers. Inductive step: Pick an arbitrary k and l such that k < l but x k > x l. When is this inversion counted? k, l n/2 : xk, x l A, counted in r A. k, l n/2 : xk, x l B, counted in r B. k n/2, l n/2 :
23 Counting Inversions: Correctness of Sort-and-Count Prove by induction. Strategy: every inversion in the data is counted exactly once. Base case: n = 1. Inductive hypothesis: Algorithm counts number of inversions correctly for all sets of n 1 or fewer numbers. Inductive step: Pick an arbitrary k and l such that k < l but x k > x l. When is this inversion counted? k, l n/2 : xk, x l A, counted in r A. k, l n/2 : xk, x l B, counted in r B. k n/2, l n/2 : xk A, x l B. Is this inversion counted by Merge-and-Count?
24 Counting Inversions: Correctness of Sort-and-Count Prove by induction. Strategy: every inversion in the data is counted exactly once. Base case: n = 1. Inductive hypothesis: Algorithm counts number of inversions correctly for all sets of n 1 or fewer numbers. Inductive step: Pick an arbitrary k and l such that k < l but x k > x l. When is this inversion counted? k, l n/2 : xk, x l A, counted in r A. k, l n/2 : xk, x l B, counted in r B. k n/2, l n/2 : xk A, x l B. Is this inversion counted by Merge-and-Count? Yes, when x l is appended to the output.
25 Computational Geometry Algorithms for geometric objects: points, lines, segments, triangles, spheres, polyhedra, ldots. Started in 1975 by Shamos and Hoey. Problems studied have applications in a vast number of fields: ecology, molecular biology, statistics, computational finance, computer graphics, computer vision,...
26 Computational Geometry Algorithms for geometric objects: points, lines, segments, triangles, spheres, polyhedra, ldots. Started in 1975 by Shamos and Hoey. Problems studied have applications in a vast number of fields: ecology, molecular biology, statistics, computational finance, computer graphics, computer vision,... Closest Pair of Points INSTANCE: A set P of n points in the plane SOLUTION: The pair of points in P that are the closest to each other.
27 Computational Geometry Algorithms for geometric objects: points, lines, segments, triangles, spheres, polyhedra, ldots. Started in 1975 by Shamos and Hoey. Problems studied have applications in a vast number of fields: ecology, molecular biology, statistics, computational finance, computer graphics, computer vision,... Closest Pair of Points INSTANCE: A set P of n points in the plane SOLUTION: The pair of points in P that are the closest to each other. At first glance, it seems any algorithm must take Ω(n 2 ) time. Shamos and Hoey figured out an ingenious O(n log n) divide and conquer algorithm.
28 Closest Pair: Set-up Let P = {p 1, p 2,..., p n } with p i = (x i, y i ). Use d(p i, p j ) to denote the Euclidean distance between p i and p j. For a specific pair of points, can compute d(p i, p j ) in O(1) time. Goal: find the pair of points p i and p j that minimise d(p i, p j ).
29 Closest Pair: Set-up Let P = {p 1, p 2,..., p n } with p i = (x i, y i ). Use d(p i, p j ) to denote the Euclidean distance between p i and p j. For a specific pair of points, can compute d(p i, p j ) in O(1) time. Goal: find the pair of points p i and p j that minimise d(p i, p j ). How do we solve the problem in 1D?
30 Closest Pair: Set-up Let P = {p 1, p 2,..., p n } with p i = (x i, y i ). Use d(p i, p j ) to denote the Euclidean distance between p i and p j. For a specific pair of points, can compute d(p i, p j ) in O(1) time. Goal: find the pair of points p i and p j that minimise d(p i, p j ). How do we solve the problem in 1D? Sort: closest pair must be adjacent in the sorted order.
31 Closest Pair: Set-up Let P = {p 1, p 2,..., p n } with p i = (x i, y i ). Use d(p i, p j ) to denote the Euclidean distance between p i and p j. For a specific pair of points, can compute d(p i, p j ) in O(1) time. Goal: find the pair of points p i and p j that minimise d(p i, p j ). How do we solve the problem in 1D? Sort: closest pair must be adjacent in the sorted order. Divide and conquer after sorting:
32 Closest Pair: Set-up Let P = {p 1, p 2,..., p n } with p i = (x i, y i ). Use d(p i, p j ) to denote the Euclidean distance between p i and p j. For a specific pair of points, can compute d(p i, p j ) in O(1) time. Goal: find the pair of points p i and p j that minimise d(p i, p j ). How do we solve the problem in 1D? Sort: closest pair must be adjacent in the sorted order. Divide and conquer after sorting: closest pair must be closest of 1. closest pair in left half: distance δ l. 2. closest pair in right half: distance δ r. 3. closest among pairs that span the left and right halves and are at most min(δ l, δ r ) apart. How many such pairs do we need to consider?
33 Closest Pair: Set-up Let P = {p 1, p 2,..., p n } with p i = (x i, y i ). Use d(p i, p j ) to denote the Euclidean distance between p i and p j. For a specific pair of points, can compute d(p i, p j ) in O(1) time. Goal: find the pair of points p i and p j that minimise d(p i, p j ). How do we solve the problem in 1D? Sort: closest pair must be adjacent in the sorted order. Divide and conquer after sorting: closest pair must be closest of 1. closest pair in left half: distance δ l. 2. closest pair in right half: distance δ r. 3. closest among pairs that span the left and right halves and are at most min(δ l, δ r ) apart. How many such pairs do we need to consider? Just one!
34 Closest Pair: Set-up Let P = {p 1, p 2,..., p n } with p i = (x i, y i ). Use d(p i, p j ) to denote the Euclidean distance between p i and p j. For a specific pair of points, can compute d(p i, p j ) in O(1) time. Goal: find the pair of points p i and p j that minimise d(p i, p j ). How do we solve the problem in 1D? Sort: closest pair must be adjacent in the sorted order. Divide and conquer after sorting: closest pair must be closest of 1. closest pair in left half: distance δ l. 2. closest pair in right half: distance δ r. 3. closest among pairs that span the left and right halves and are at most min(δ l, δ r ) apart. How many such pairs do we need to consider? Just one! Generalize the second idea to 2D.
35 Closest Pair: Algorithm Skeleton 1. Divide P into two sets Q and R of n/2 points such that each point in Q has x-coordinate less than any point in R. 2. Recursively compute closest pair in Q and in R, respectively.
36 Closest Pair: Algorithm Skeleton 1. Divide P into two sets Q and R of n/2 points such that each point in Q has x-coordinate less than any point in R. 2. Recursively compute closest pair in Q and in R, respectively. 3. Let δ 1 be the distance computed for Q, δ 2 be the distance computed for R, and δ = min(δ 1, δ2). 4. Compute pair (q, r) of points such that q Q, r R, d(q, r) < δ and d(q, r) is the smallest possible.
37 Closest Pair: Algorithm Skeleton 1. Divide P into two sets Q and R of n/2 points such that each point in Q has x-coordinate less than any point in R. 2. Recursively compute closest pair in Q and in R, respectively. 3. Let δ 1 be the distance computed for Q, δ 2 be the distance computed for R, and δ = min(δ 1, δ2). 4. Compute pair (q, r) of points such that q Q, r R, d(q, r) < δ and d(q, r) is the smallest possible. Sketch of proof of correctness by induction: Of the two points in the closest pair (i) both are in Q: computed correctly by recursive call. (ii) both are in R: computed correctly by recursive call. (iii) one is in Q and the other is in R: computed correctly in O(n) time by the procedure we will discuss. Strategy: Pairs of points for which we do not compute the distance between cannot be the closest pair. Overall running time is O(n log n).
38 Closest Pair: Conquer Step Line L passes through right-most point in Q.
39 Closest Pair: Conquer Step Line L passes through right-most point in Q. Let S be the set of points within distance δ of L. Claim: There exist q Q, r R such that d(q, r) < δ if and only if there exist s, s S such that d(s, s ) < δ.
40 Closest Pair: Packing Argument Intuition: if there are too many points in S that are closer than δ to each other, then there must be a pair in Q or in R that are less than δ apart.
41 Closest Pair: Packing Argument Intuition: if there are too many points in S that are closer than δ to each other, then there must be a pair in Q or in R that are less than δ apart. Let S y denote the set of points in S sorted by increasing y-coordinate and let s y denote the y-coordinate of a point s S.
42 Closest Pair: Packing Argument Intuition: if there are too many points in S that are closer than δ to each other, then there must be a pair in Q or in R that are less than δ apart. Let S y denote the set of points in S sorted by increasing y-coordinate and let s y denote the y-coordinate of a point s S. Claim: If there exist s, s S such that d(s, s ) < δ then s and s are at most 15 indices apart in S y.
43 Closest Pair: Packing Argument Intuition: if there are too many points in S that are closer than δ to each other, then there must be a pair in Q or in R that are less than δ apart. Let S y denote the set of points in S sorted by increasing y-coordinate and let s y denote the y-coordinate of a point s S. Claim: If there exist s, s S such that d(s, s ) < δ then s and s are at most 15 indices apart in S y. Converse of the claim: If there exist s, s S such that s appears 16 or more indices after s in S y, then s y s y δ.
44 Closest Pair: Packing Argument Intuition: if there are too many points in S that are closer than δ to each other, then there must be a pair in Q or in R that are less than δ apart. Let S y denote the set of points in S sorted by increasing y-coordinate and let s y denote the y-coordinate of a point s S. Claim: If there exist s, s S such that d(s, s ) < δ then s and s are at most 15 indices apart in S y. Converse of the claim: If there exist s, s S such that s appears 16 or more indices after s in S y, then s y s y δ. Use the claim in an algorithm: For every point s S y, compute distances only to the next 15 points in S y. Other pairs of points cannot be candidates for the closest pair.
45 Closest Pair: Proof of Packing Argument Claim: If there exist s, s S such that s appears 16 or more indices after s in S y, then s y s y δ. Pack the plane with squares of side δ/2.
46 Closest Pair: Proof of Packing Argument Claim: If there exist s, s S such that s appears 16 or more indices after s in S y, then s y s y δ. Pack the plane with squares of side δ/2. Each square contains at most one point.
47 Closest Pair: Proof of Packing Argument Claim: If there exist s, s S such that s appears 16 or more indices after s in S y, then s y s y δ. Pack the plane with squares of side δ/2. Each square contains at most one point. Let s lie in one of the squares in the first row.
48 Closest Pair: Proof of Packing Argument Claim: If there exist s, s S such that s appears 16 or more indices after s in S y, then s y s y δ. Pack the plane with squares of side δ/2. Each square contains at most one point. Let s lie in one of the squares in the first row. Any point in the fourth row has a y-coordinate at least δ more than s y.
49 Closest Pair: Final Algorithm
50 Closest Pair: Final Algorithm
51 Closest Pair: Final Algorithm
52 Integer Multiplication Multiply Integers INSTANCE: Two n-digit binary integers x and y SOLUTION: The product xy
53 Integer Multiplication Multiply Integers INSTANCE: Two n-digit binary integers x and y SOLUTION: The product xy Multiply two n-digit integers.
54 Integer Multiplication Multiply Integers INSTANCE: Two n-digit binary integers x and y SOLUTION: The product xy Multiply two n-digit integers. Result has at most 2n digits.
55 Integer Multiplication Multiply Integers INSTANCE: Two n-digit binary integers x and y SOLUTION: The product xy Multiply two n-digit integers. Result has at most 2n digits. Algorithm we learnt in school takes
56 Integer Multiplication Multiply Integers INSTANCE: Two n-digit binary integers x and y SOLUTION: The product xy Multiply two n-digit integers. Result has at most 2n digits. Algorithm we learnt in school takes O(n 2 ) operations. Size of the input is not 2 but 2n,
57 Divide-and-Conquer Algorithm Assume integers are binary. Let us use divide and conquer
58 Divide-and-Conquer Algorithm Assume integers are binary. Let us use divide and conquer by splitting each number into first n/2 bits and last n/2 bits. Let x be split into x 0 (lower-order bits) and x 1 (higher-order bits) and y into y 0 (lower-order bits) and y 1 (higher-order bits). xy =
59 Divide-and-Conquer Algorithm Assume integers are binary. Let us use divide and conquer by splitting each number into first n/2 bits and last n/2 bits. Let x be split into x 0 (lower-order bits) and x 1 (higher-order bits) and y into y 0 (lower-order bits) and y 1 (higher-order bits). xy = (x 1 2 n/2 + x 0 )(y 1 2 n/2 + y 0 ) = x 1 y 1 2 n + (x 1 y 0 + x 0 y 1 )2 n/2 + x 0 y 0.
60 Divide-and-Conquer Algorithm Assume integers are binary. Let us use divide and conquer by splitting each number into first n/2 bits and last n/2 bits. Let x be split into x 0 (lower-order bits) and x 1 (higher-order bits) and y into y 0 (lower-order bits) and y 1 (higher-order bits). xy = (x 1 2 n/2 + x 0 )(y 1 2 n/2 + y 0 ) = x 1 y 1 2 n + (x 1 y 0 + x 0 y 1 )2 n/2 + x 0 y 0. Algorithm: each of x 1, x 0, y 1, y 0 has n/2 bits, so we can compute x 1 y 1, x 1 y 0, x 0 y 1, and x 0 y 0 recursively, and merge the answers in O(n) time.
61 Divide-and-Conquer Algorithm Assume integers are binary. Let us use divide and conquer by splitting each number into first n/2 bits and last n/2 bits. Let x be split into x 0 (lower-order bits) and x 1 (higher-order bits) and y into y 0 (lower-order bits) and y 1 (higher-order bits). xy = (x 1 2 n/2 + x 0 )(y 1 2 n/2 + y 0 ) = x 1 y 1 2 n + (x 1 y 0 + x 0 y 1 )2 n/2 + x 0 y 0. Algorithm: each of x 1, x 0, y 1, y 0 has n/2 bits, so we can compute x 1 y 1, x 1 y 0, x 0 y 1, and x 0 y 0 recursively, and merge the answers in O(n) time. What is the running time T (n)?
62 Divide-and-Conquer Algorithm Assume integers are binary. Let us use divide and conquer by splitting each number into first n/2 bits and last n/2 bits. Let x be split into x 0 (lower-order bits) and x 1 (higher-order bits) and y into y 0 (lower-order bits) and y 1 (higher-order bits). xy = (x 1 2 n/2 + x 0 )(y 1 2 n/2 + y 0 ) = x 1 y 1 2 n + (x 1 y 0 + x 0 y 1 )2 n/2 + x 0 y 0. Algorithm: each of x 1, x 0, y 1, y 0 has n/2 bits, so we can compute x 1 y 1, x 1 y 0, x 0 y 1, and x 0 y 0 recursively, and merge the answers in O(n) time. What is the running time T (n)? T (n) 4T (n/2) + cn
63 Divide-and-Conquer Algorithm Assume integers are binary. Let us use divide and conquer by splitting each number into first n/2 bits and last n/2 bits. Let x be split into x 0 (lower-order bits) and x 1 (higher-order bits) and y into y 0 (lower-order bits) and y 1 (higher-order bits). xy = (x 1 2 n/2 + x 0 )(y 1 2 n/2 + y 0 ) = x 1 y 1 2 n + (x 1 y 0 + x 0 y 1 )2 n/2 + x 0 y 0. Algorithm: each of x 1, x 0, y 1, y 0 has n/2 bits, so we can compute x 1 y 1, x 1 y 0, x 0 y 1, and x 0 y 0 recursively, and merge the answers in O(n) time. What is the running time T (n)? T (n) 4T (n/2) + cn O(n 2 )
64 Improving the Algorithm Four sub-problems lead to an O(n 2 ) algorithm. How can we reduce the number of sub-problems?
65 Improving the Algorithm Four sub-problems lead to an O(n 2 ) algorithm. How can we reduce the number of sub-problems? We do not need to compute x1 y 0 and x 0 y 1 independently; we just need their sum.
66 Improving the Algorithm Four sub-problems lead to an O(n 2 ) algorithm. How can we reduce the number of sub-problems? We do not need to compute x1 y 0 and x 0 y 1 independently; we just need their sum. x1 y 1 + (x 1 y 0 + x 0 y 1 ) + x 0 y 0 = (x 0 + x 1 )(y 0 + y 1 ) Compute x 1 y 1, x 0 y 0 and (x 0 + x 1 )(y 0 + y 1 ) recursively and then compute (x 1 y 0 + x 0 y 1 ) by subtraction. We have three sub-problems of size n/2. Strategy: simple arithmetic manipulations. What is the running time T (n)?
67 Improving the Algorithm Four sub-problems lead to an O(n 2 ) algorithm. How can we reduce the number of sub-problems? We do not need to compute x1 y 0 and x 0 y 1 independently; we just need their sum. x1 y 1 + (x 1 y 0 + x 0 y 1 ) + x 0 y 0 = (x 0 + x 1 )(y 0 + y 1 ) Compute x 1 y 1, x 0 y 0 and (x 0 + x 1 )(y 0 + y 1 ) recursively and then compute (x 1 y 0 + x 0 y 1 ) by subtraction. We have three sub-problems of size n/2. Strategy: simple arithmetic manipulations. What is the running time T (n)? T (n) 3T (n/2) + cn
68 Improving the Algorithm Four sub-problems lead to an O(n 2 ) algorithm. How can we reduce the number of sub-problems? We do not need to compute x1 y 0 and x 0 y 1 independently; we just need their sum. x1 y 1 + (x 1 y 0 + x 0 y 1 ) + x 0 y 0 = (x 0 + x 1 )(y 0 + y 1 ) Compute x 1 y 1, x 0 y 0 and (x 0 + x 1 )(y 0 + y 1 ) recursively and then compute (x 1 y 0 + x 0 y 1 ) by subtraction. We have three sub-problems of size n/2. Strategy: simple arithmetic manipulations. What is the running time T (n)? T (n) 3T (n/2) + cn O(n log 2 3 ) = O(n 1.59 )
69 Final Algorithm
5.4 Closest Pair of Points
5.4 Closest Pair of Points Closest Pair of Points Closest pair. Given n points in the plane, find a pair with smallest Euclidean distance between them. Fundamental geometric primitive. Graphics, computer
Closest Pair Problem
Closest Pair Problem Given n points in d-dimensions, find two whose mutual distance is smallest. Fundamental problem in many applications as well as a key step in many algorithms. p q A naive algorithm
Computational Geometry. Lecture 1: Introduction and Convex Hulls
Lecture 1: Introduction and convex hulls 1 Geometry: points, lines,... Plane (two-dimensional), R 2 Space (three-dimensional), R 3 Space (higher-dimensional), R d A point in the plane, 3-dimensional space,
Closest Pair of Points. Kleinberg and Tardos Section 5.4
Closest Pair of Points Kleinberg and Tardos Section 5.4 Closest Pair of Points Closest pair. Given n points in the plane, find a pair with smallest Euclidean distance between them. Fundamental geometric
SECTION 10-2 Mathematical Induction
73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms
Pigeonhole Principle Solutions
Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such
CSE 421 Algorithms: Divide and Conquer
CSE 421 Algorithms: Divide and Conquer Summer 2011 Larry Ruzzo Thanks to Paul Beame, James Lee, Kevin Wayne for some slides hw2 empirical run times e n Plot Time vs n Fit curve to it (e.g., with Excel)
Many algorithms, particularly divide and conquer algorithms, have time complexities which are naturally
Recurrence Relations Many algorithms, particularly divide and conquer algorithms, have time complexities which are naturally modeled by recurrence relations. A recurrence relation is an equation which
28 Closest-Point Problems -------------------------------------------------------------------
28 Closest-Point Problems ------------------------------------------------------------------- Geometric problems involving points on the plane usually involve implicit or explicit treatment of distances
3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
CS473 - Algorithms I
CS473 - Algorithms I Lecture 4 The Divide-and-Conquer Design Paradigm View in slide-show mode 1 Reminder: Merge Sort Input array A sort this half sort this half Divide Conquer merge two sorted halves Combine
CSE373: Data Structures and Algorithms Lecture 3: Math Review; Algorithm Analysis. Linda Shapiro Winter 2015
CSE373: Data Structures and Algorithms Lecture 3: Math Review; Algorithm Analysis Linda Shapiro Today Registration should be done. Homework 1 due 11:59 pm next Wednesday, January 14 Review math essential
3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
Lecture 3: Finding integer solutions to systems of linear equations
Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture
ALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite
ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,
Catalan Numbers. Thomas A. Dowling, Department of Mathematics, Ohio State Uni- versity.
7 Catalan Numbers Thomas A. Dowling, Department of Mathematics, Ohio State Uni- Author: versity. Prerequisites: The prerequisites for this chapter are recursive definitions, basic counting principles,
Introduction to Algorithms March 10, 2004 Massachusetts Institute of Technology Professors Erik Demaine and Shafi Goldwasser Quiz 1.
Introduction to Algorithms March 10, 2004 Massachusetts Institute of Technology 6.046J/18.410J Professors Erik Demaine and Shafi Goldwasser Quiz 1 Quiz 1 Do not open this quiz booklet until you are directed
DESARGUES THEOREM DONALD ROBERTSON
DESARGUES THEOREM DONALD ROBERTSON Two triangles ABC and A B C are said to be in perspective axially when no two vertices are equal and when the three intersection points AC A C, AB A B and BC B C are
God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886)
Chapter 2 Numbers God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886) God created the integers and the rest is the work
PYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
Scan-Line Fill. Scan-Line Algorithm. Sort by scan line Fill each span vertex order generated by vertex list
Scan-Line Fill Can also fill by maintaining a data structure of all intersections of polygons with scan lines Sort by scan line Fill each span vertex order generated by vertex list desired order Scan-Line
MATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
Algorithms. Margaret M. Fleck. 18 October 2010
Algorithms Margaret M. Fleck 18 October 2010 These notes cover how to analyze the running time of algorithms (sections 3.1, 3.3, 4.4, and 7.1 of Rosen). 1 Introduction The main reason for studying big-o
Solving Geometric Problems with the Rotating Calipers *
Solving Geometric Problems with the Rotating Calipers * Godfried Toussaint School of Computer Science McGill University Montreal, Quebec, Canada ABSTRACT Shamos [1] recently showed that the diameter of
8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
Today. Binary addition Representing negative numbers. Andrew H. Fagg: Embedded Real- Time Systems: Binary Arithmetic
Today Binary addition Representing negative numbers 2 Binary Addition Consider the following binary numbers: 0 0 1 0 0 1 1 0 0 0 1 0 1 0 1 1 How do we add these numbers? 3 Binary Addition 0 0 1 0 0 1 1
6.2 Permutations continued
6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of
a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
The program also provides supplemental modules on topics in geometry and probability and statistics.
Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students
csci 210: Data Structures Recursion
csci 210: Data Structures Recursion Summary Topics recursion overview simple examples Sierpinski gasket Hanoi towers Blob check READING: GT textbook chapter 3.5 Recursion In general, a method of defining
Random Map Generator v1.0 User s Guide
Random Map Generator v1.0 User s Guide Jonathan Teutenberg 2003 1 Map Generation Overview...4 1.1 Command Line...4 1.2 Operation Flow...4 2 Map Initialisation...5 2.1 Initialisation Parameters...5 -w xxxxxxx...5
MATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.
MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P
Well-Separated Pair Decomposition for the Unit-disk Graph Metric and its Applications
Well-Separated Pair Decomposition for the Unit-disk Graph Metric and its Applications Jie Gao Department of Computer Science Stanford University Joint work with Li Zhang Systems Research Center Hewlett-Packard
BALTIC OLYMPIAD IN INFORMATICS Stockholm, April 18-22, 2009 Page 1 of?? ENG rectangle. Rectangle
Page 1 of?? ENG rectangle Rectangle Spoiler Solution of SQUARE For start, let s solve a similar looking easier task: find the area of the largest square. All we have to do is pick two points A and B and
Solving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
Euclidean Minimum Spanning Trees Based on Well Separated Pair Decompositions Chaojun Li. Advised by: Dave Mount. May 22, 2014
Euclidean Minimum Spanning Trees Based on Well Separated Pair Decompositions Chaojun Li Advised by: Dave Mount May 22, 2014 1 INTRODUCTION In this report we consider the implementation of an efficient
= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
Geometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
IV. ALGEBRAIC CONCEPTS
IV. ALGEBRAIC CONCEPTS Algebra is the language of mathematics. Much of the observable world can be characterized as having patterned regularity where a change in one quantity results in changes in other
A Note on Maximum Independent Sets in Rectangle Intersection Graphs
A Note on Maximum Independent Sets in Rectangle Intersection Graphs Timothy M. Chan School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1, Canada tmchan@uwaterloo.ca September 12,
RN-Codings: New Insights and Some Applications
RN-Codings: New Insights and Some Applications Abstract During any composite computation there is a constant need for rounding intermediate results before they can participate in further processing. Recently
Georgia Standards of Excellence Curriculum Map. Mathematics. GSE 8 th Grade
Georgia Standards of Excellence Curriculum Map Mathematics GSE 8 th Grade These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. GSE Eighth Grade
This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
with functions, expressions and equations which follow in units 3 and 4.
Grade 8 Overview View unit yearlong overview here The unit design was created in line with the areas of focus for grade 8 Mathematics as identified by the Common Core State Standards and the PARCC Model
2.1 Complexity Classes
15-859(M): Randomized Algorithms Lecturer: Shuchi Chawla Topic: Complexity classes, Identity checking Date: September 15, 2004 Scribe: Andrew Gilpin 2.1 Complexity Classes In this lecture we will look
We can express this in decimal notation (in contrast to the underline notation we have been using) as follows: 9081 + 900b + 90c = 9001 + 100c + 10b
In this session, we ll learn how to solve problems related to place value. This is one of the fundamental concepts in arithmetic, something every elementary and middle school mathematics teacher should
Mathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
Solutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
Kapitel 1 Multiplication of Long Integers (Faster than Long Multiplication)
Kapitel 1 Multiplication of Long Integers (Faster than Long Multiplication) Arno Eigenwillig und Kurt Mehlhorn An algorithm for multiplication of integers is taught already in primary school: To multiply
Chapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling
Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one
Binary Multiplication
Binary Multiplication Q: How do we multiply two numbers? eg. 12, 345 6, 789 111105 987600 8641500 + 74070000 83, 810, 205 10111 10101 10111 00000 1011100 0000000 + 101110000 111100011 Pad, multiply and
DATA MINING CLUSTER ANALYSIS: BASIC CONCEPTS
DATA MINING CLUSTER ANALYSIS: BASIC CONCEPTS 1 AND ALGORITHMS Chiara Renso KDD-LAB ISTI- CNR, Pisa, Italy WHAT IS CLUSTER ANALYSIS? Finding groups of objects such that the objects in a group will be similar
Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.
Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of
discuss how to describe points, lines and planes in 3 space.
Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position
Algorithm Design and Analysis Homework #1 Due: 5pm, Friday, October 4, 2013 TA email: ada@csie.ntu.edu.tw. === Homework submission instructions ===
Algorithm Design and Analysis Homework #1 Due: 5pm, Friday, October 4, 2013 TA email: ada@csie.ntu.edu.tw === Homework submission instructions === For Problem 1, commit your source code and a brief documentation
Converting a Number from Decimal to Binary
Converting a Number from Decimal to Binary Convert nonnegative integer in decimal format (base 10) into equivalent binary number (base 2) Rightmost bit of x Remainder of x after division by two Recursive
IMPROVING PERFORMANCE OF RANDOMIZED SIGNATURE SORT USING HASHING AND BITWISE OPERATORS
Volume 2, No. 3, March 2011 Journal of Global Research in Computer Science RESEARCH PAPER Available Online at www.jgrcs.info IMPROVING PERFORMANCE OF RANDOMIZED SIGNATURE SORT USING HASHING AND BITWISE
A Non-Linear Schema Theorem for Genetic Algorithms
A Non-Linear Schema Theorem for Genetic Algorithms William A Greene Computer Science Department University of New Orleans New Orleans, LA 70148 bill@csunoedu 504-280-6755 Abstract We generalize Holland
Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
KEANSBURG SCHOOL DISTRICT KEANSBURG HIGH SCHOOL Mathematics Department. HSPA 10 Curriculum. September 2007
KEANSBURG HIGH SCHOOL Mathematics Department HSPA 10 Curriculum September 2007 Written by: Karen Egan Mathematics Supervisor: Ann Gagliardi 7 days Sample and Display Data (Chapter 1 pp. 4-47) Surveys and
096 Professional Readiness Examination (Mathematics)
096 Professional Readiness Examination (Mathematics) Effective after October 1, 2013 MI-SG-FLD096M-02 TABLE OF CONTENTS PART 1: General Information About the MTTC Program and Test Preparation OVERVIEW
I. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
2009 Chicago Area All-Star Math Team Tryouts Solutions
1. 2009 Chicago Area All-Star Math Team Tryouts Solutions If a car sells for q 1000 and the salesman earns q% = q/100, he earns 10q 2. He earns an additional 100 per car, and he sells p cars, so his total
Linear Codes. Chapter 3. 3.1 Basics
Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length
The Tower of Hanoi. Recursion Solution. Recursive Function. Time Complexity. Recursive Thinking. Why Recursion? n! = n* (n-1)!
The Tower of Hanoi Recursion Solution recursion recursion recursion Recursive Thinking: ignore everything but the bottom disk. 1 2 Recursive Function Time Complexity Hanoi (n, src, dest, temp): If (n >
To define function and introduce operations on the set of functions. To investigate which of the field properties hold in the set of functions
Chapter 7 Functions This unit defines and investigates functions as algebraic objects. First, we define functions and discuss various means of representing them. Then we introduce operations on functions
z 0 and y even had the form
Gaussian Integers The concepts of divisibility, primality and factoring are actually more general than the discussion so far. For the moment, we have been working in the integers, which we denote by Z
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation
Medical Information Management & Mining. You Chen Jan,15, 2013 You.chen@vanderbilt.edu
Medical Information Management & Mining You Chen Jan,15, 2013 You.chen@vanderbilt.edu 1 Trees Building Materials Trees cannot be used to build a house directly. How can we transform trees to building materials?
Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
CSC148 Lecture 8. Algorithm Analysis Binary Search Sorting
CSC148 Lecture 8 Algorithm Analysis Binary Search Sorting Algorithm Analysis Recall definition of Big Oh: We say a function f(n) is O(g(n)) if there exists positive constants c,b such that f(n)
#1-12: Write the first 4 terms of the sequence. (Assume n begins with 1.)
Section 9.1: Sequences #1-12: Write the first 4 terms of the sequence. (Assume n begins with 1.) 1) a n = 3n a 1 = 3*1 = 3 a 2 = 3*2 = 6 a 3 = 3*3 = 9 a 4 = 3*4 = 12 3) a n = 3n 5 Answer: 3,6,9,12 a 1
! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.
Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three
Divide-and-conquer algorithms
Chapter 2 Divide-and-conquer algorithms The divide-and-conquer strategy solves a problem by: 1 Breaking it into subproblems that are themselves smaller instances of the same type of problem 2 Recursively
2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
DEFINITION 5.1.1 A complex number is a matrix of the form. x y. , y x
Chapter 5 COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the field C of complex numbers is via the arithmetic of matrices. DEFINITION 5.1.1 A complex number is a matrix of
Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.
Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material
Homework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
Sample Induction Proofs
Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
Regions in a circle. 7 points 57 regions
Regions in a circle 1 point 1 region points regions 3 points 4 regions 4 points 8 regions 5 points 16 regions The question is, what is the next picture? How many regions will 6 points give? There's an
Full and Complete Binary Trees
Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full
The Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned
Approximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
Section 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
1. The volume of the object below is 186 cm 3. Calculate the Length of x. (a) 3.1 cm (b) 2.5 cm (c) 1.75 cm (d) 1.25 cm
Volume and Surface Area On the provincial exam students will need to use the formulas for volume and surface area of geometric solids to solve problems. These problems will not simply ask, Find the volume
DELAWARE MATHEMATICS CONTENT STANDARDS GRADES 9-10. PAGE(S) WHERE TAUGHT (If submission is not a book, cite appropriate location(s))
Prentice Hall University of Chicago School Mathematics Project: Advanced Algebra 2002 Delaware Mathematics Content Standards (Grades 9-10) STANDARD #1 Students will develop their ability to SOLVE PROBLEMS
CALCULATIONS & STATISTICS
CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents
2) Write in detail the issues in the design of code generator.
COMPUTER SCIENCE AND ENGINEERING VI SEM CSE Principles of Compiler Design Unit-IV Question and answers UNIT IV CODE GENERATION 9 Issues in the design of code generator The target machine Runtime Storage
CONTENTS 1. Peter Kahn. Spring 2007
CONTENTS 1 MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 2 The Integers 1 2.1 The basic construction.......................... 1 2.2 Adding integers..............................
Answer: (a) Since we cannot repeat men on the committee, and the order we select them in does not matter, ( )
1. (Chapter 1 supplementary, problem 7): There are 12 men at a dance. (a) In how many ways can eight of them be selected to form a cleanup crew? (b) How many ways are there to pair off eight women at the
CS 103X: Discrete Structures Homework Assignment 3 Solutions
CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering
The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.
The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,
α = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
The Set Data Model CHAPTER 7. 7.1 What This Chapter Is About
CHAPTER 7 The Set Data Model The set is the most fundamental data model of mathematics. Every concept in mathematics, from trees to real numbers, is expressible as a special kind of set. In this book,
Voyager Sopris Learning Vmath, Levels C-I, correlated to the South Carolina College- and Career-Ready Standards for Mathematics, Grades 2-8
Page 1 of 35 VMath, Level C Grade 2 Mathematical Process Standards 1. Make sense of problems and persevere in solving them. Module 3: Lesson 4: 156-159 Module 4: Lesson 7: 220-223 2. Reason both contextually
Basics of Counting. The product rule. Product rule example. 22C:19, Chapter 6 Hantao Zhang. Sample question. Total is 18 * 325 = 5850
Basics of Counting 22C:19, Chapter 6 Hantao Zhang 1 The product rule Also called the multiplication rule If there are n 1 ways to do task 1, and n 2 ways to do task 2 Then there are n 1 n 2 ways to do