Binomial Coefficient Identities and Encoding/Decoding
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1 Binomial Coefficient Identities and Encoding/Decoding CSE21 Winter 2017, Day 18 (B00), Day 12 (A00) February 24,
2 MT2 Review Sessions Today and Tomorrow! TODAY 2/24 Solis pm TOMORROW 2/25 Center pm Have you done the PPs? Read the TTK?
3 Recall: Fixed-density Binary Strings Rosen p. 413 How many length nbinary strings contain k ones? Density is number of ones Objects: all strings made up of 0 1, 0 2, 1 1, 1 2, 1 3, 1 4 n! Categories: strings that agree except subscripts Size of each category: k!(n-k)! # categories = (# objects) / (size of each category) =n! / ( k! (n-k)! ) =C(n,k) =
4 Binomial Coefficient Identities What's an identity? An equation that is always true. To prove LHS = RHS Use algebraic manipulations of formulas. OR Interpret each side as counting some collection of strings, and then prove a statements about those sets of strings.
5 Symmetry Identity Theorem: Rosen p. 411
6 Symmetry Identity Theorem: Rosen p. 411 Proof 1: Use formula
7 Symmetry Identity Theorem: Rosen p. 411 Proof 1: Use formula Proof 2: Combinatorial interpretation? LHS counts number of binary strings of length n with k ones RHS counts number of binary strings of length n with n-k ones
8 Symmetry Identity Theorem: Rosen p. 411 Proof 1: Use formula Proof 2: Combinatorial interpretation? LHS counts number of binary strings of length n with k ones and n-k zeros RHS counts number of binary strings of length n with n-k ones and k zeros
9 Symmetry Identity Theorem: Rosen p. 411 Proof 1: Use formula Proof 2: Combinatorial interpretation? LHS counts number of binary strings of length n with k ones and n-k zeros RHS counts number of binary strings of length n with n-k ones and k zeros Can match up these two sets by pairing each string with another where 0s, 1s are flipped. This bijection means the two sets have the same size. So LHS = RHS.
10 Pascal's Identity Theorem: Rosen p. 418 Proof 1: Use formula Proof 2: Combinatorial interpretation? LHS counts number of binary strings??? RHS counts number of binary strings???
11 Pascal's Identity Theorem: Rosen p. 418 Proof 2: Combinatorial interpretation? LHS counts number of binary strings of length n+1 that have k ones. RHS counts number of binary strings??? Length n+1 binary strings with k ones
12 Pascal's Identity Theorem: Rosen p. 418 Proof 2: Combinatorial interpretation? LHS counts number of binary strings of length n+1 that have k ones. RHS counts number of binary strings??? Start with 1 Start with 0
13 Pascal's Identity How many length n+1 strings start with 1 and have k ones in total? Rosen p. 418 A. C(n+1, k+1) B. C(n, k) C. C(n, k+1) D. C(n, k-1) E. None of the above. Start with 1 Start with 0
14 Pascal's Identity How many length n+1 strings start with 0 and have k ones in total? Rosen p. 418 A. C(n+1, k+1) B. C(n, k) C. C(n, k+1) D. C(n, k-1) E. None of the above. Start with 1 Start with 0
15 Pascal's Identity Theorem: Rosen p. 418 Proof 2: Combinatorial interpretation? LHS counts number of binary strings of length n+1 that have k ones. RHS counts number of binary strings of length n+1 that have k ones, using two cases Start with 1 Start with 0
16 Sum Identity Theorem: Rosen p. 417 What set does the LHS count? A. Binary strings of length n that have k ones. B. Binary strings of length n that start with 1. C. Binary strings of length n that have any number of ones. D. None of the above.
17 Sum Identity Theorem: Rosen p. 417 Proof : Combinatorial interpretation? LHS counts number of binary strings of length n that have any number of 1s. By sum rule, we can break up the set of binary strings of length n into disjoint sets based on how many 1s they have, then add their sizes. RHS counts number of binary strings of length n. This is the same set so LHS = RHS.
18 Encoding Fixed-density Binary Strings Rosen p. 413 What's the smallest number of bits that we need to specify a binary string of length n if we know it hask ones andn-k zeros? A. n B. k C. log 2 ( C(n,k) ) D. n-k E. None of the above
19 Data Compression Store / transmit information in as little space as possible
20 Data Compression: Video Video: stored as sequence of still frames. Idea: instead of storing each frame fully, record change from previous frame.
21 Data Compression: Run-Length Encoding Image: described as grid of pixels, each with RED, GREEN, BLUE values. Idea: instead of storing RGB value of each pixel, store run-length of run of same color. When is this a good coding mechanism? Will there be any loss in this compression?
22 Lossy Compression: Singular Value Decomposition Image: described as grid of pixels, each with RED, GREEN, BLUE values. Idea: use Linear Algebra to compress data to a fraction of its size, with minimal loss.
23 Data Compression: Trade-off Complicated compression scheme save storage space may take a long time to encode / decode Data Encoding Algorithm Decoding Algorithm Data Stored in Computer
24 Encoding: Binary Palindromes Palindrome: string that reads the same forward and backward. How many length n binary palindromes are there? A. 2 n B. n C. n/2 D. log 2 n E. None of the above
25 Encoding: Binary Palindromes Palindrome: string that reads the same forward and backward. How many bits are (optimally) required to encode a length n binary palindrome? A. n B. n-1 C. n/2 D. log 2 n/2 Is there an algorithm that achieves this?
26 Encoding: Fixed Density Strings Goal: encode a length n binary string that we know has k ones (and n-k zeros). How would you represent such a string with n-1bits?
27 Encoding: Fixed Density Strings Goal: encode a length n binary string that we know has k ones (and n-k zeros). How would you represent such a string with n-1bits? Can we do better? What if we know k is much less than n?
28 Encoding: Fixed Density Strings Goal: encode a length n binary string that we know has k ones (and n-k zeros). How would you represent such a string with n-1bits? Can we do better? What if we know k is only slightly less than n?
29 Encoding: Fixed Density Strings Goal: encode a length n binary string that we know has k ones (and n-k zeros). How would you represent such a string with n-1bits? Can we do better? What if k is about half of n? Idea: give positions of 1s in the string within some smaller window. -Fix window size. -If there is a 1 in the current "window" in the string, record its position and move the window over. -Otherwise, record a 0 and move the window over.
30 Encoding: Fixed Density Strings Idea: give positions of 1s in the string within some smaller window. -Fix window size. -If there is a 1 in the current "window" in the string, record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ?
31 Encoding: Fixed Density Strings Idea: give positions of 1s in the string within some smaller window. -Fix window size. -If there is a 1 in the current "window" in the string, record its position and move the window over. -Otherwise, record a 0 and move the window over. This window has several 1s. Example n=12, k=3, window size n/k = 4. Use the first! How do we encode s = ? There's a 1! What's its position? Output:
32 Encoding: Fixed Density Strings Idea: give positions of 1s in the string within some smaller window. -Fix window size. -If there is a 1 in the current "window" in the string, record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: 01 There's a 1! What's its position?
33 Encoding: Fixed Density Strings Idea: give positions of 1s in the string within some smaller window. -Fix window size. -If there is a 1 in the current "window" in the string, record its position and move the window over. -Otherwise, record a 0 and move the window Slide over. the window to start just after the Example n=12, k=3, window size n/k = 4. part of the string we have recorded. How do we encode s = ? Output: 01 There's a 1! What's its position?
34 Encoding: Fixed Density Strings Idea: give positions of 1s in the string within some smaller window. -Fix window size. -If there is a 1 in the current "window" in the string, record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: 0100 There's a 1! What's its position?
35 Encoding: Fixed Density Strings Idea: give positions of 1s in the string within some smaller window. -Fix window size. -If there is a 1 in the current "window" in the string, record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: 0100 No 1s in this window.
36 Encoding: Fixed Density Strings Idea: give positions of 1s in the string within some smaller window. -Fix window size. -If there is a 1 in the current "window" in the string, record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: No 1s in this window.
37 Encoding: Fixed Density Strings Idea: give positions of 1s in the string within some smaller window. -Fix window size. -If there is a 1 in the current "window" in the string, record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: There's a 1! What's its position?
38 Encoding: Fixed Density Strings Idea: give positions of 1s in the string within some smaller window. -Fix window size. -If there is a 1 in the current "window" in the string, record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: There's a 1! What's its position?
39 Encoding: Fixed Density Strings Idea: give positions of 1s in the string within some smaller window. -Fix window size. -If there is a 1 in the current "window" in the string, record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: No 1s in this window.
40 Encoding: Fixed Density Strings Idea: give positions of 1s in the string within some smaller window. -Fix window size. -If there is a 1 in the current "window" in the string, record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: No 1s in this window.
41 Encoding: Fixed Density Strings Idea: give positions of 1s in the string within some smaller window. -Fix window size. -If there is a 1 in the current "window" in the string, record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: Compressed to 8 bits! But can we recover the original string? Decoding
42 Encoding: Fixed Density Strings With n=12, k=3, window size n/k = 4. Output: Can be parsed as the (intended) input: s = ? But also: 01: one in position 1 0: no ones 00: one in position 0 11: one in position 3 0: no ones s' = Problem: two different inputs with same output. Can't uniquely decode.
43 Compression Algorithm A valid compression algorithm must: -Have outputs of shorter (or same) length as input. - Be uniquely decodable.
44 Encoding: Fixed Density Strings Can we modify this algorithm to get unique decodability? Idea: use marker bit to indicate when to interpret output as a position. -Fix window size. -If there is a 1 in the current "window" in the string, record a 1 to interpret next bits as position, then record its position and move the window over. -Otherwise, record a 0 and move the window over.
45 Encoding: Fixed Density Strings Idea: use marker bit to indicate when to interpret output as a position. -Fix window size. -If there is a 1 in the current "window" in the string, record a 1 to interpret next bits as position, then record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output:
46 Encoding: Fixed Density Strings Idea: use marker bit to indicate when to interpret output as a position. -Fix window size. -If there is a 1 in the current "window" in the string, record a 1 to interpret next bits as position, then record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output:
47 Encoding: Fixed Density Strings Idea: use marker bit to indicate when to interpret output as a position. -Fix window size. -If there is a 1 in the current "window" in the string, record a 1 to interpret next bits as position, then record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: What output corresponds to these first few bits? A. 0 C. 01 E. None of the above. B. 1 D. 101
48 Encoding: Fixed Density Strings Idea: use marker bit to indicate when to interpret output as a position. -Fix window size. -If there is a 1 in the current "window" in the string, record a 1 to interpret next bits as position, then record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: 101 Interpret next bits as position of 1; this position is 01
49 Encoding: Fixed Density Strings Idea: use marker bit to indicate when to interpret output as a position. -Fix window size. -If there is a 1 in the current "window" in the string, record a 1 to interpret next bits as position, then record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: 101
50 Encoding: Fixed Density Strings Idea: use marker bit to indicate when to interpret output as a position. -Fix window size. -If there is a 1 in the current "window" in the string, record a 1 to interpret next bits as position, then record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: 101 A C B D
51 Encoding: Fixed Density Strings Idea: use marker bit to indicate when to interpret output as a position. -Fix window size. -If there is a 1 in the current "window" in the string, record a 1 to interpret next bits as position, then record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: Interpret next bits as position of 1; this position is 00
52 Encoding: Fixed Density Strings Idea: use marker bit to indicate when to interpret output as a position. -Fix window size. -If there is a 1 in the current "window" in the string, record a 1 to interpret next bits as position, then record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output:
53 Encoding: Fixed Density Strings Idea: use marker bit to indicate when to interpret output as a position. -Fix window size. -If there is a 1 in the current "window" in the string, record a 1 to interpret next bits as position, then record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: No 1s in this window.
54 Encoding: Fixed Density Strings Idea: use marker bit to indicate when to interpret output as a position. -Fix window size. -If there is a 1 in the current "window" in the string, record a 1 to interpret next bits as position, then record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output:
55 Encoding: Fixed Density Strings Idea: use marker bit to indicate when to interpret output as a position. -Fix window size. -If there is a 1 in the current "window" in the string, record a 1 to interpret next bits as position, then record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: Interpret next bits as position of 1; this position is 11
56 Encoding: Fixed Density Strings Idea: use marker bit to indicate when to interpret output as a position. -Fix window size. -If there is a 1 in the current "window" in the string, record a 1 to interpret next bits as position, then record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output:
57 Encoding: Fixed Density Strings Idea: use marker bit to indicate when to interpret output as a position. -Fix window size. -If there is a 1 in the current "window" in the string, record a 1 to interpret next bits as position, then record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: No 1s in this window.
58 Encoding: Fixed Density Strings Idea: use marker bit to indicate when to interpret output as a position. -Fix window size. -If there is a 1 in the current "window" in the string, record a 1 to interpret next bits as position, then record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: Compare to previous output: Output uses more bits than last time. Any redundancies?
59 Encoding: Fixed Density Strings Idea: use marker bit to indicate when to interpret output as a position. -Fix window size. -If there is a 1 in the current "window" in the string, record a 1 to interpret next bits as position, then record its position and move the window over. -Otherwise, record a 0 and move the window over. Example n=12, k=3, window size n/k = 4. How do we encode s = ? Output: Compare to previous output: * Since k is known, after we see the last 1, we can stop since the rest are 0s. *
60 Encoding: Fixed Density Strings procedure WindowEncode (input: b 1 b 2 b n, with exactly k ones and n-k zeros) 1. w := floor (n/k) 2. count := 0 3. location := 1 4. While count < k: 5. If there is a 1 in the window starting at current location 6. Output 1 as a marker, then output position of first 1 in window. 7. Increment count. 8. Update location to immediately after first 1 in this window. 9. Else 10. Output Update location to next index after current window. Uniquely decodable?
61 Decoding: Fixed Density Strings procedure WindowDecode (input: x 1 x 2 x m, target is exactly k ones and n-k zeros) 1. w := floor ( n/k ) 2. b := floor ( log 2 (w)) 3. s := empty string 4. i := 0 5. While i < m 6. If x i = 0 7. s += 0 0 (j times) 8. i += 1 9. Else 10. p := decimal value of the bits x i+1 x i+b 11. s += 0 0 (p times) 12. s += i := i+b If length(s) < n 15. s += 0 0 ( n-length(s) times ) 16. Output s.
62 Encoding/Decoding: Fixed Density Strings Correctness? E(s) = result of encoding string s of length n with k 1s, using WindowEncode. D(t) = result of decoding string t to create a string of length n with k 1s, using WindowDecode. Well-defined functions? Inverses? Can show that for each s, D(E(s)) = s. Proof uses Strong Induction!
63 Encoding/Decoding: Fixed Density Strings Output size? Assume n/k is a power of two. Consider s a binary string of length n with k 1s. How long is E(s)? A. n-1 B. log 2 (n/k) C. Depends on where 1s are located in s. D. None of the above.
64 Encoding/Decoding: Fixed Density Strings Output size? Assume n/k is a power of two. Consider s a binary string of length n with k 1s. For which strings is E(s) shortest? A. More 1s toward the beginning. B. More 1s toward the end. C. 1s spread about evenly throughout.
65 Encoding/Decoding: Fixed Density Strings Output size? Assume n/k is a power of two. Consider s a binary string of length n with k 1s. Best case : 1s toward the beginning of the string. E(s) has -One bit for each 1 in s to indicate that next bits denote positions in window. -log 2 (n/k) bits for each 1 in s to specify position of that 1 in a window. -k such ones. -No bits representing empty windows because all 0s are either "caught" in windows with 1s or after the last 1. Total E(s) = k log 2 (n/k) + k
66 Encoding/Decoding: Fixed Density Strings Output size? Assume n/k is a power of two. Consider s a binary string of length n with k 1s. Worst case : 1s toward the end of the string. E(s) has -Some bits for empty windows since there are no 1s in first several windows. -One bit for each 1 in s to indicate that next bits denote positions in window. -log 2 (n/k) bits for each 1 in s to specify position of that 1 in a window. -k such ones. What's an upper bound on the number of these bits? A. n C. 1 B. n-k D. k
67 Encoding/Decoding: Fixed Density Strings Output size? Assume n/k is a power of two. Consider s a binary string of length n with k 1s. Worst case : 1s toward the end of the string. E(s) has -At most k bits for empty windows since at most k nonoverlapping windows of length n/k will fit in a string of length n. -One bit for each 1 in s to indicate that next bits denote positions in window. -log 2 (n/k) bits for each 1 in s to specify position of that 1 in a window. -k such ones. Total E(s) <= k log 2 (n/k) + 2k
68 Encoding/Decoding: Fixed Density Strings Output size? Assume n/k is a power of two. Consider s a binary string of length n with k 1s. k log 2 (n/k) + k <= E(s) <= k log 2 (n/k) + 2k Using this inequality, there are at most length n strings with k 1s. A. 2 n D. (n/k) k B. n E. None of the above. C. (n/k) 2
69 Encoding/Decoding: Fixed Density Strings Output size? Assume n/k is a power of two. Consider s a binary string of length n with k 1s. Given E(s) <= k log 2 (n/k) + 2k, we need at most k log 2 (n/k) + 2k bits to represent all length n binary strings with k 1s. Hence, there are at most 2 many such strings.
70 Encoding/Decoding: Fixed Density Strings Output size? Assume n/k is a power of two. Consider s a binary string of length n with k 1s. Given E(s) <= k log 2 (n/k) + 2k, we need at most k log 2 (n/k) + 2k bits to represent all length n binary strings with k 1s. Hence, there are at most 2 many such strings.
71 Encoding/Decoding: Fixed Density Strings Output size? Assume n/k is a power of two. Consider s a binary string of length n with k 1s. Given E(s) <= k log 2 (n/k) + 2k, we need at most k log 2 (n/k) + 2k bits to represent all length n binary strings with k 1s. Hence, there are at most 2 many such strings. C(n,k) = # Length n binary strings with k 1s <= (4n/k) k
72 Bounds for Binomial Coefficients Using windowencode(): Lower bound? Idea: find a way to count a subset of the fixed density binary strings.. Some fixed density binary strings have one 1 in each of k chunks of size n/k. How many such strings are there? A. n n B. k! C. (n/k) k D. C(n,k) k E. None of the above.
73 Bounds for Binomial Coefficients Using windowencode(): Using evenly spread strings: Counting helps us analyze our compression algorithm. Compression algorithms help us count.
74 Announcements MT REVIEW TODAY Solis pm TOMORROW Center pm Have you done the PPs? Read the TTK?
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