Algorithm Engineering (aka. How to Write Fast Code) CS26 S260 – Lecture cture 6 Yan n Gu I/O Algorithms and Parallel Samplesort
Review of Samplesort CS260: Algorithm Semisort Engineering Lecture 6 Course Policy 2
Sample-sort outline Analo logou gous s to mult ltiw iway ay quic ickso ksort 1. 1. Sp Spli lit in input ut array in into 𝑂 contiguo iguous us suba barra rrays ys of siz ize 𝑂 . So Sort subar arrays rays recursi sivel vely … 𝑂 , sorted 𝑂
Sample-sort outline Analo logou gous s to mult ltiw iway ay quic ickso ksort 𝑂 , sorted 1. 1. Sp Spli lit in input ut array in into 𝑂 contiguo iguous us suba barra rrays ys of siz ize 𝑂 . So Sort subar arrays rays recursi sivel vely y (sequ equent entia ially lly) …
Sample-sort outline 2. 2. Choo oose se 𝑂 − 1 “good” pivots 𝑂 , sorted 𝑞 1 ≤ 𝑞 2 ≤ ⋯ ≤ 𝑞 𝑂−1 3. 3. Dis istribu ribute te su subar barrays rays in into o buckets ckets , , ac accordin ording g to … pivot vots Size ≈ 𝑂 ≤ 𝑞 1 ≤ ≤ 𝑞 2 ≤ ⋯ ≤ 𝑞 𝑂−1 ≤ Bucket 1 Bucket 2 Bucket 𝑂
Sample-sort outline 4. Recurs 4. cursively ively sort rt the buckets ckets ≤ 𝑞 1 ≤ ≤ 𝑞 2 ≤ ⋯ ≤ 𝑞 𝑂−1 ≤ Bucket 1 Bucket 2 Bucket 𝑂 5. 5. Copy py conca oncatenated tenated buckets ckets bac ack k to input put ar arra ray sorted
Review of Samplesort CS260: Algorithm Semisort Engineering Lecture 6 Course Policy 7
What is semisort? key 45 12 45 61 28 61 61 45 28 45 Value 2 5 3 9 5 9 8 1 7 5 • Input: • An array of records with associated keys • Assume keys can be hashed to the range [𝑜 𝑙 ] • Goal: • All records with equal keys should be adjacent
What is semisort? key 12 61 61 61 45 45 45 45 28 28 Value 5 8 9 9 2 5 1 3 7 5 • Input: • An array of records with associated keys • Assume keys can be hashed to the range [𝑜 𝑙 ] • Goal: • All records with equal keys should be adjacent
What is semisort? key 45 45 45 45 12 61 61 61 28 28 Value 2 5 1 3 5 8 9 9 7 5 • Input: • An array of records with associated keys • Assume keys can be hashed to the range [𝑜 𝑙 ] • Goal: • All records with equal keys should be adjacent • Different keys are not necessarily sorted • Records with equal keys do not need to be sorted by their values
What is semisort? key 45 45 45 45 12 61 61 61 28 28 Value 1 5 3 2 5 8 9 9 7 5 • Input: • An array of records with associated keys • Assume keys can be hashed to the range [𝑜 𝑙 ] • Goal: • All records with equal keys should be adjacent • Different keys are not necessarily sorted • Records with equal keys do not need to be sorted by their values
Semisort is one of the most useful primitives in parallel algorithms Parallel In-Place Algorithms: Theory and Practice Julienne: A Framework for Parallel Graph Algorithms using Work- efficient Bucketing Semi-Asymmetric Parallel Graph Algorithms for NVRAMs Efficient BVH Construction via Approximate Agglomerative Clustering Theoretically-Efficient and Practical Parallel DBSCAN 12
Why is semisort so useful? (albeit not seen before) • Semisorting can be done by sorting, but faster (less restriction) • Theoretically can be done in 𝑃 𝑜 work not 𝑃 𝑜 log 𝑜 work • Can be used to implement counting / integer sort • Integer sort: given 𝑜 key-value pairs with keys in range [1, … , 𝑜] , query the KV-pairs with a certain key • Counting sort: given 𝑜 key-value pairs with keys in range [1, … , 𝑜] , query the number of KV-pairs with a certain key • In database community, this is called the GroupBy operator 13
Why is semisort so useful? (albeit not seen before) • Semisorting can be done by sorting, but faster (less restriction) • Theoretically can be done in 𝑃 𝑜 work not 𝑃 𝑜 log 𝑜 work • Can be used to implement counting / integer sort keys 37 … 58 … 92 … 92 56 key value Linked 12 8 11 lists of values 9 19 56 52 14
Attempts – Sequentially: Pre-allocated array 92 56 keys 37 … 58 … 92 … key value Arrays 12 8 11 of values 9 44 19 52 31 56 Problem Need to pre-count the number of each key
Another use case for semisrot • Generate adjacency array for a graph Sorted Edge list edge list (3,5) (3,5) 2 6 (1,7) (3,7) (2,3) (3,6) 3 4 (3,6) (5,4) 1 5 (5,4) (1,6) 7 (3,7) (1,7) (1,6) (2,3)
What is semisort? key 45 45 45 45 12 61 61 61 28 28 Value 1 5 3 2 5 8 9 9 7 5 • Input: • An array of records with associated keys • Assume keys can be hashed to the range [𝑜 𝑙 ] • Goal: • All records with equal keys should be adjacent • Different keys are not necessarily sorted • Records with equal keys do not need to be sorted by their values
Why is semisort hard? key 45 45 45 45 12 61 61 61 28 28 Value 1 5 3 2 5 8 9 9 7 5 • There can be many duplicate keys • Heavy keys • Or, there can be almost no duplicate keys • Light keys
Implement integer sort using semisort key 45 45 45 45 12 61 61 61 28 28 Value 1 5 3 2 5 8 9 9 7 5 • Input: 𝒐 KV-pairs with key in [𝑜] • Step 1: hash the keys (i.e., for 𝒍 𝒋 , 𝒘 𝒋 , generate 𝒊 𝒋 = 𝐢𝐛𝐭𝐢(𝒍 𝒋 ) ) • Step 2: semisort 𝒊 𝒋 , (𝒍 𝒋 , 𝒘 𝒋 ) , and resolve conflicts • Step 3: get the pointer for each key 𝒍 𝒋
The Top-Down Parallel Semisort Algorithm 22
The main goal estimate key counts • And tell the heavy keys from light ones. By how? Sampling! • For a key appear more than 𝐨/𝒖 times, we call it a heavy key • Otherwise, we call it a light key • We can treat them separately
The algorithm • Take 𝒖 log 𝒐 samples and sort them • For those keys with more than log 𝒐 appearances, we mark them as heavy keys, others are light keys • We give each heavy key a bucket, and the another 𝒖 buckets for light keys each corresponds to a range of 𝒐 𝒍 /𝒖 • The input keys are hashed into 𝒐 𝒍 • In total we have no more than 2𝑢 buckets • The rest of the algorithm is pretty similar to samplesort
Phase 1: Sampling and sorting 1. Select a sample set 𝑇 with 𝑢 log 𝑜 of keys 2. Sort 𝑇 …… Sampling …… S Sorting 17 17 …… 5 5 5 8 8 8 8 8 11 17 (Counting)
Phase 2: Bucket Construction Sorted samples: 17 17 …… 5 5 5 8 8 8 8 8 11 17 Counting & Filtering Light keys Heavy keys Range 0-15 16-31 keys 65 … 8 20 keys 5 11 17 21 26 31 ...
At the end of Phase 2 • In total we have no more than 2𝑢 buckets • 𝑢 of them are for light keys • Then we construct a hash table for the heavy keys • Now we know which bucket each KV-pair (𝒍 𝒋 , 𝒘 𝒋 ) goes to: • If 𝑙 𝑗 is found in the hash table, assign it to the associated heavy bucket • Otherwise, it goes to the light bucket based on the range of 𝑙 𝑗 • The rest of the algorithm is almost identical to samplesort
Sample-sort outline Analogous to multiway quicksort 𝑂/𝑢 1. Split input array into 𝑂 contiguous subarrays of size 𝑂 𝑢 … 𝑂 𝑂
Sample-sort outline Analogous to multiway quicksort Size ≈ 𝑢 1. Split input array into 𝑂/𝑢 contiguous subarrays of size 𝑢 . Sort subarrays recursively (sequentially) …
Sample-sort outline 2. Distribute subarrays into buckets … … ≤ 𝑞 1 ≤ ≤ 𝑞 2 ≤ ⋯ ≤ 𝑞 𝑂−1 ≤ Bucket 1 Bucket 2 Bucket 𝑂
Sample-sort outline Only for the light buckets 3. Recursively sort the buckets … Bucket 1 Bucket 2 Bucket 𝑂 4. Copy concatenated buckets back to input array sorted
Difference 2: subarrays are not sorted • For simplicity, assume 𝒐 = 𝟐𝟕 , and the input is [𝟐, 𝟑, 𝟒, 𝟓, 𝟐, 𝟐, 𝟒, 𝟒, 𝟐, 𝟑, 𝟑, 𝟓, 𝟐, 𝟑, 𝟓, 𝟓] • First, get the count for each subarray in each bucket [𝟐, 𝟐, 𝟐, 𝟐, 𝟑, 𝟏, 𝟑, 𝟏, 𝟐, 𝟑, 𝟏, 𝟐, 𝟐, 𝟐, 𝟏, 𝟑] • Then, transpose the array and scan to compute the offsets [𝟐, 𝟑, 𝟐, 𝟐, 𝟐, 𝟏, 𝟑, 𝟐, 𝟐, 𝟑, 𝟏, 𝟏, 𝟐, 𝟏, 𝟐, 𝟑] [𝟏, 𝟐, 𝟒, 𝟓, 𝟔, 𝟕, 𝟕, 𝟗, 𝟘, 𝟐𝟏, 𝟐𝟑, 𝟐𝟑, 𝟐𝟑, 𝟐𝟒, 𝟐𝟒, 𝟐𝟓] • Lastly, move each element to the corresponding bucket [∅, ∅, ∅, ∅, ∅, ∅, ∅, ∅, ∅, ∅, ∅, ∅, ∅, ∅, ∅, ∅] [𝟐, ∅, ∅, ∅, ∅, 𝟑, ∅, ∅, ∅, 𝟒, ∅, ∅, 𝟓, ∅, ∅, ∅] [𝟐, 𝟐, 𝟐, ∅, ∅, 𝟑, ∅, ∅, ∅, 𝟒, 𝟒, 𝟒, 𝟓, ∅, ∅, ∅] 32
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