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- 1. I/O-efficient Algorithms and Data Structures Lars Arge Professor and center director Aarhus University ALMADA summer school August 5, 2013
- 2. Lars Arge 2 I/O-Model • Parameters N = # elements in problem instance B = # elements that fits in disk block M = # elements that fits in main memory T = # output size in searching problem • We often assume that M>B2 • I/O: Movement of block between memory and disk D P M Block I/O I/O-efficient Algorithms and Data Structures
- 3. Lars Arge 3 Fundamental Bounds Internal External • Scanning: N • Sorting: N log N • Permuting • Searching: • Note: – Linear I/O: O(N/B) – Permuting not linear – Permuting and sorting bounds are equal in all practical cases – B factor VERY important: – Cannot sort optimally with search tree NBlog B N B N B Mlog B N NB N B N B N B Mlog }log,min{ B N B N B MNN N2log I/O-efficient Algorithms and Data Structures
- 4. Lars Arge 4 External Sort • Merge sort: – Create N/M memory sized sorted runs – Merge runs together M/B at a time phases using I/Os each • Distribution sort similar (but harder – split elements) )( B NO)(log M N B MO I/O-efficient Algorithms and Data Structures
- 5. Lower bounds • Permuting N elements according to a given permutation takes I/Os in “indivisibility” model • Sorting N elements takes I/Os in comparison model – Proved using counting arguments Lars Arge 5 I/O-efficient Algorithms and Data Structures })log,(min{ B N B N B MN )log( B N B N B M !)!())(( NBN B N B M t
- 6. Lars Arge 6 Cache-Oblivious Model • N, B, and M as in I/O-model – Assumed that M>B2 (tall cache assumption) • M and B not used in algorithm description • Block transfers by optimal paging strategy Analyze in two-level model Efficient on one level, efficient on all levels (of fully associative hierarchy using LRU) I/O-efficient Algorithms and Data Structures
- 7. Lars Arge 7 Cache-Oblivious Distribution Sort • Break into subarrays of size – Recursively sort each subarray • Distribute into buckets of size ~ – Recursively sort each bucket • Distribution performed in I/Os = N I/O-efficient Algorithms and Data Structures )log( B N B N B MO N N N MNO MNONTN NT B N B N if)( if)()(2 )( )( B N O
- 8. Lars Arge 8 Sorting summary • External merge or distribution sort takes I/Os – Merge-sort based on M/B-way merging – Distribution sort based on -way distribution and (complicated) split elements finding – Also cache-oblivious using -way distribution • Optimal in comparison model • lower bound in stronger indivisibility model – Holds even for permuting )log( B N B N B MO })log,(min{ B N B N B MN B M I/O-efficient Algorithms and Data Structures N
- 9. Distribution sweeping • Combination of distribution and plane sweep – Used to solve batched geometric problems • General idea: – Divide plane into M/B slabs with O(N/(M/B)) endpoints each – Sweep plane top-down while finding solutions involving objects in different slabs – Distribute data to M/B slabs recourse in each slab • Examples: – Orthogonal line segment intersection – Rectangle intersection – Homework Lars Arge 9 I/O-efficient Algorithms and Data Structures
- 10. Outline • Thursday: – Sorting upper bound (merge- and distribution-sort) – Sorting (permuting) lower bound – Cache-oblivious sorting – Batched geometric problems: Distribution sweeping • Today: – Searching (B-trees) – Batched searching (Buffer-trees) – I/O-efficient priority queues – I/O-efficient list ranking Lars Arge 10 I/O-efficient Algorithms and Data Structures
- 11. Lars Arge 11 – If nodes stored arbitrarily on disk Search in I/Os Rangesearch in I/Os • Binary search tree: – Standard method for search among N elements – We assume elements in leaves – Search traces at least one root-leaf path External Search Trees )(log2 NO )(log2 N )(log2 TNO I/O-efficient Algorithms and Data Structures
- 12. Lars Arge 12 External Search Trees • BFS blocking: – Block height – Output elements blocked Rangesearch in I/Os • Optimal: O(N/B) space and query )(log2 B )(B )(log)(log/)(log 22 NOBONO B )(log B T B N )(log B T B N I/O-efficient Algorithms and Data Structures
- 13. Lars Arge 13 Cache-Oblivious Search Tree • “Van Emde Boas” recursive layout: NBlog B1 B A B2 N Nlog2 1 Nlog2 1 Recursive memory layout: A B1 B2 BN I/O-efficient Algorithms and Data Structures
- 14. Lars Arge 14 Cache-Oblivious Search Tree • Search analysis: – Conceptually stop recursion when recursive subtrees have size and each recursive subtree fits in a block NBlog – Search path “hits” blocks)(log)(log/log NOBN BB B B B B B B )(logB I/O-efficient Algorithms and Data Structures
- 15. Lars Arge 15 • Maintaining BFS blocking during updates? – Balance normally maintained in search trees using rotations • Seems very difficult to maintain BFS blocking during rotation – Also need to make sure output (leaves) is blocked! Dynamic External Search Trees? x y x y I/O-efficient Algorithms and Data Structures
- 16. Lars Arge 16 B-trees • BFS-blocking naturally corresponds to tree with fan-out • B-trees balanced by allowing node degree to vary – Rebalancing performed by splitting and merging nodes )(B I/O-efficient Algorithms and Data Structures
- 17. Lars Arge 17 • (a,b)-tree uses linear space and has height Choosing a,b = each node/leaf stored in one disk block space and query (a,b)-tree • T is an (a,b)-tree (a≥2 and b≥2a-1) – All leaves on the same level and contain between a and b elements – Except for the root, all nodes have degree between a and b – Root has degree between 2 and b )(log NO a )(log B T B N )(B tree I/O-efficient Algorithms and Data Structures
- 18. Lars Arge 18 (a,b)-Tree Insert • Insert: Search and insert element in leaf v DO v has b+1 elements/children Split v: make nodes v’ and v’’ with and elements insert element (ref) in parent(v) (make new root if necessary) v=parent(v) • Insert touch nodes bb 2 1 ab 2 1 )(log Na v v’ v’’ 2 1b 2 1b 1b I/O-efficient Algorithms and Data Structures
- 19. Lars Arge 19 (2,4)-Tree Insert I/O-efficient Algorithms and Data Structures
- 20. 20 (a,b)-Tree Delete • Delete: Search and delete element from leaf v DO v has a-1 elements/children Fuse v with sibling v’: move children of v’ to v delete element (ref) from parent(v) (delete root if necessary) If v has >b (and ≤ a+b-1<2b) children split v v=parent(v) • Delete touch nodes)(log NO a v v 1a 12a I/O-efficient Algorithms and Data Structures Lars Arge
- 21. Lars Arge 21 (2,4)-Tree Delete I/O-efficient Algorithms and Data Structures
- 22. Lars Arge 22 • (a,b)-tree properties: – If b=2a-1 every update can cause many rebalancing operations – If b≥2a update only cause O(1) rebalancing operations amortized – If b>2a only rebalancing operations amortized * Both somewhat hard to show – If b=4a easy to show that update causes rebalance operations amortized * After split during insert a leaf contains 4a/2=2a elements * After fuse during delete a leaf contains between 2a and 5a elements (split if more than 3a between 3/2a and 5/2a) (a,b)-Tree )()( 11 2 aa OO b )log(1 NO aa insert delete (2,3)-tree I/O-efficient Algorithms and Data Structures
- 23. Lars Arge 23 B-tree Summary • B-trees: (a,b)-trees with a,b = – O(N/B) space – O(logB N+T/B) query – O(logB N) update • B-trees with elements in the leaves sometimes called B+-tree • Construction in I/Os – Sort elements and construct leaves – Build tree level-by-level bottom-up )(B )log( B N B N B MO I/O-efficient Algorithms and Data Structures
- 24. Lars Arge 24 Generalized B-tree • B-tree with branching parameter b and leaf parameter k (b,k≥8) – All leaves on same level and contain between 1/4k and k elements – Except for the root, all nodes have degree between 1/4b and b – Root has degree between 2 and b • B-tree with leaf parameter – O(N/B) space – Height – amortized leaf rebalance operations – amortized internal node rebalance operations )(log B N bO )(1 kO )log( 1 B N bkb O )(Bk I/O-efficient Algorithms and Data Structures
- 25. B-tree Sorting • Normally we can sort N elements in O(N log N) time with search tree – Insert all elements one-by-one (construct tree) – Output in sorted order using in-order traversal • Same algorithm using B-tree use I/Os – A factor of non-optimal • We would like to have dynamic data structure to use in algorithms I/O operations )log( NNO B )( log log B B M BO )log( B N BMB N O )log( 1 B N BMB O I/O-efficient Algorithms and Data Structures 25Lars Arge
- 26. 26 • Main idea: Logically group nodes together and add buffers – Insertions done in a “lazy” way – elements inserted in buffers – When a buffer runs full elements are pushed one level down – Buffer-emptying in O(M/B) I/Os every block touched constant number of times on each level inserting N elements (N/B blocks) costs I/Os)log( B N BMB N O Buffer-tree Technique B B M elements fan-out M/B )(log B N BMO I/O-efficient Algorithms and Data Structures Lars Arge
- 27. Lars Arge 27 • Definition: – B-tree with branching parameter and leaf parameter B – Size M buffer in each internal node • Updates: – Add time-stamp to insert/delete element – Collect B elements in memory before inserting in root buffer – Perform buffer-emptying when buffer runs full Basic Buffer-tree B M $m$ blocks M B M B M ...4 1 B I/O-efficient Algorithms and Data Structures
- 28. Lars Arge 28 Basic Buffer-tree • Note: – Buffer can be larger than M during recursive buffer-emptying * Elements distributed in sorted order at most M elements in buffer unsorted – Rebalancing needed when “leaf-node” buffer emptied * Leaf-node buffer-emptying only performed after all full internal node buffers are emptied $m$ blocks M B M B M ...4 1 B I/O-efficient Algorithms and Data Structures
- 29. Lars Arge 29 Basic Buffer-tree • Internal node buffer-empty: – Load first M (unsorted) elements into memory and sort them – Merge elements in memory with rest of (already sorted) elements – Scan through sorted list while * Removing “matching” insert/deletes * Distribute elements to child buffers – Recursively empty full child buffers • Emptying buffer of size X takes O(X/B+M/B)=O(X/B) I/Os $m$ blocks M B M B M ...4 1 I/O-efficient Algorithms and Data Structures
- 30. Lars Arge 30 Basic Buffer-tree • Buffer-empty of leaf node with K elements in leaves – Sort buffer as previously – Merge buffer elements with elements in leaves – Remove “matching” insert/deletes obtaining K’ elements – If K’<K then * Add K-K’ “dummy” elements and insert in “dummy” leaves Otherwise * Place K elements in leaves * Repeatedly insert block of elements in leaves and rebalance • Delete dummy leaves and rebalance when all full buffers emptied K I/O-efficient Algorithms and Data Structures
- 31. Lars Arge 31 Basic Buffer-tree • Invariant: Buffers of nodes on path from root to emptied leaf-node are empty • Insert rebalancing (splits) performed as in normal B-tree • Delete rebalancing: v’ buffer emptied before fuse of v – Necessary buffer emptyings performed before next dummy- block delete – Invariant maintained v vv’ v v’ v’’ I/O-efficient Algorithms and Data Structures
- 32. Lars Arge 32 Basic Buffer-tree • Analysis: – Not counting rebalancing, a buffer-emptying of node with X ≥ M elements (full) takes O(X/B) I/Os total full node emptying cost I/Os – Delete rebalancing buffer-emptying (non-full) takes O(M/B) I/Os cost of one split/fuse O(M/B) I/Os – During N updates * O(N/B) leaf split/fuse * internal node split/fuse Total cost of N operations: I/Os )log( B N B M B M B N O )log( B N B N B MO )log( B N B N B MO I/O-efficient Algorithms and Data Structures
- 33. Lars Arge 33 Basic Buffer-tree • Emptying all buffers after N insertions: Perform buffer-emptying on all nodes in BFS-order resulting full-buffer emptyings cost I/Os empty non-full buffers using O(M/B) O(N/B) I/Os • N elements can be sorted using buffer tree in I/Os )log( B N B N B MO )( B M B N O $m$ blocks M B M B M ...4 1 B )log( B N B N B MO I/O-efficient Algorithms and Data Structures
- 34. Lars Arge 34 • Batching of operations on B-tree using M-sized buffers – I/O updates amortized – All buffers emptied in I/Os • One-dim. rangesearch operations can also be supported in I/Os amortized – Search elements handle lazily like updates – All elements in relevant sub-trees reported during buffer-emptying – Buffer-emptying in O(X/B+T’/B), where T’ is reported elements Buffer-tree Summary )log( 1 B N BMB O )log( 1 B T B N BMB O $m$ blocks )log( B N B N B MO I/O-efficient Algorithms and Data Structures
- 35. Lars Arge 35 Orthogonal Line Segment Intersection • Sweep plane top-down while maintaining search tree T on vertical segments crossing sweep line (by x-coordinates) – Top endpoint of vertical segment: Insert in T – Bottom endpoint of vertical segment: Delete from T – Horizontal segment: Perform range query with x-interval on T I/Os using buffer-tree I/O-efficient Algorithms and Data Structures )log( B T B N B N B MO
- 36. Lars Arge 36 • Basic buffer tree can be used in external priority queue • To delete minimal element: – Empty all buffers on leftmost path – Delete elements in leftmost leaf and keep in memory – Deletion of next M minimal elements free – Inserted elements checked against minimal elements in memory • I/Os every O(M) delete amortized Buffered Priority Queue )log( B N BMB M O M4 1 )log( 1 B N BMB O )( B M B I/O-efficient Algorithms and Data Structures
- 37. Lars Arge 37 List Ranking • Problem: – Given N-vertex linked list stored in array – Compute rank (number in list) of each vertex • One of the simplest graph problem one can think of • Straightforward O(N) internal algorithm – Also uses O(N) I/Os in external memory • Much harder to get external algorithm 3 4 5 9 68 27 101 5 2 6 73 4 108 9 )log( B N BMB N O I/O-efficient Algorithms and Data Structures
- 38. Lars Arge 38 List Ranking • We will solve more general problem: – Given N-vertex linked list with edge-weights stored in array – Compute sum of weights (rank) from start for each vertex • List ranking: All edge weights one • Note: Weight stored in array entry together with edge (next vertex) 1 5 2 6 73 4 108 9 1 1 11 1 1 1 1 1 1 I/O-efficient Algorithms and Data Structures
- 39. Lars Arge 39 List Ranking • Algorithm: 1. Find and mark independent set of vertices 2. “Bridge-out” independent set: Add new edges 3. Recursively rank resulting list 4. “Bridge-in” independent set: Compute rank of independent set • Step 1, 2 and 4 in I/Os • Independent set of size αN for 0 < α ≤ 1 I/Os 11 111 1 1 11 1 2 2 2 1 3 4 6 8 9 102 5 7 )log( B N BMB N O )log()log())1(()( B N BMB N B N BMB N OONTNT I/O-efficient Algorithms and Data Structures
- 40. Lars Arge 40 List Ranking: Bridge-out/in • Obtain information (edge or rang) of successor – Make copy of original list – Sort original list by successor id – Scan original and copy together to obtain successor information – Sort modified original list by id I/Os 11 2 3 4 5 9 68 27 102 3 4 95 86 7 103 4 5 9 68 27 103 4 8 9 627 10 )log( B N BMB N O I/O-efficient Algorithms and Data Structures
- 41. Lars Arge 41 List Ranking: Independent Set • Easy to design randomized algorithm: – Scan list and flip a coin for each vertex – Independent set is vertices with head and successor with tails Independent set of expected size N/4 • Deterministic algorithm: – 3-color vertices (no vertex same color as predecessor/successor) – Independent set is vertices with most popular color Independent set of size at least N/3 • 3-coloring I/O algorithm)log( B N BMB N O )log( B N BMB N O )log( B N BMB N O 3 4 5 9 68 27 10 I/O-efficient Algorithms and Data Structures
- 42. Lars Arge 42 List Ranking: 3-coloring • Algorithm: – Consider forward and backward lists (heads/tails in two lists) – Color forward lists (except tail) alternately red and blue – Color backward lists (except tail) alternately green and blue 3-coloring 3 4 5 9 68 27 10 I/O-efficient Algorithms and Data Structures
- 43. Lars Arge 43 List Ranking: Forward List Coloring • Identify heads and tails • For each head, insert red element in priority-queue (priority=position) • Repeatedly: – Extract minimal element from queue – Access and color corresponding element in list – Insert opposite color element corresponding to successor in queue • Scan of list • O(N) priority-queue operations I/Os `3 4 5 9 68 27 10 )log( B N BMB N O I/O-efficient Algorithms and Data Structures
- 44. Lars Arge 44 List Ranking Summary • Simplest graph problem: Traverse linked list • Very easy O(N) algorithm in internal memory • Much more difficult external memory – Finding independent set via 3-coloring – Bridging vertices in/out • Permuting bound best possible – Also true for other graph problems )log( B N BMB N O })log,(min{ B N B N B MNO 3 4 5 9 68 27 101 5 2 6 73 4 108 9 I/O-efficient Algorithms and Data Structures
- 45. Lars Arge 45 List Ranking Summary • External list ranking algorithm similar to PRAM algorithm – Sometimes external algorithms by “PRAM algorithm simulation” • Forward list coloring algorithm example of “time forward processing” – Use external priority-queue to send information “forward in time” to vertices to be processed later • PRAM algorithm ideas and time forward processing used in many graph algorithms 3 4 5 9 68 27 10 I/O-efficient Algorithms and Data Structures
- 46. References • External Memory Geometric Data Structures (section 1-5) Lecture notes by L. Arge • I/O-Efficient Graph Algorithms (section 1-2) Lecture notes by N. Zeh • Cache-Oblivious Algorithms. M. Frigo, C. Leiserson, H. Prokop, and S. Ramachandran. Proc. FOCS '99. Lars Arge 46 I/O-efficient Algorithms and Data Structures
- 47. Summary • We discussed – Sorting – Hardness of permuting (sorting) – Batched geometric problems – Searching: B-trees, buffer-trees, priority-queue – Graph algorithms: List-ranking – Cache-oblivious algorithms • Important techniques – Multiway merging/distribution – Distribution sweeping – Multiway-trees, buffered data structures – Time-forward processing (and PRAM simulation) Lars Arge 47 I/O-efficient Algorithms and Data Structures
- 48. Thanks large@cs.au.dk www.madalgo.au.dk

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