Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Persistent data structures 1 Ephemeral: A modification destroys the version which we modify. Persistent: Modifications are nondestructive. Each modification creates a new version. All version coexist. We have a big data structure that represent all versions 2 Partially persistent: Can access any version, but can modify only the most recent one. V1 V2 V3 V4 V5 3 fully persistent: Can access and modify any version any number of times . V1 V4 V2 V5 V3 V5 4 confluently persistent: fully persistent and there is an operation that combines two or more versions to one new version. V1 V4 V2 V5 V3 V5 5 Purely functional: You are not allowed to change a field in a node after it is initialized. This is everything you can do in a pure functional programming language 6 Example -- stacks S’’ z Two operations S’ = push(x,S) (x,S’) = pop(S) S’ x y S S3 S’ = push(x,S) S’’ = push(z,S’) (y,S3) = pop( S) S S’ Stacks are automatically fully persistent. S3 s’’ 7 Example -- queues Q3 Q3 Two operations Q’ = inject(x,Q) (x,Q’) = pop(Q) y Q’’ Q’ Q x Q z Q’ Q’’ Q’ = inject(x,Q) Q’’ = inject(z,Q’) (y,Q3) = pop( Q’’) We have partial persistent, We never want to store two different values in the same field How do we make the queue fully persistent ? 8 Example -- double ended queues Q Q’ Q’ Q four operations x Q’ = inject(x,Q) (x,Q’) = eject(Q) Q’ = push(x,Q) (x,Q’) = pop(Q) (x,Q’) = eject(Q) Q’’ = inject(z,Q’) Here its not even obvious how to get partial persistence ? 9 Maybe we should use stacks Stacks are easy. We know how to simulate queues with stacks. So we should be able to get persistent queues this way... push pop inject eject When one of the stacks gets empty we split the other 4 3 2 1 4 3 eject 2 1 eject 10 Deque by stack simulation (ephemeral analysis) = | |Sl| - |Sr| | Each operation changes the potential by O(1) The amortized cost of the reverse is 0. 4 3 2 1 4 3 eject 2 1 eject In a persistent setting it is not clear that this potential is well defined 11 Deque by stack simulation (partial persistence) = | |Sl| - |Sr| | Where S is the “live” stack, the one which we can modify Everything still works When we do the reversal in order not to modify any other stack we copy the nodes ! 4 3 2 1 4 3 eject 2 1 eject 12 Deque by stack simulation (full persistence) Can repeat the expensive operation over and over again .... eject or .... eject A sequence of n operations that costs (n2) 13 Summary so far Stacks are automatically fully persistent Got partially persistent queues in O(1) time per pop/inject Got partially persistent deques in O(1) amortized time per operation How about fully persistent queues ? Partially persistent search trees, other data structures ? Can we do something general ? 14 How about search trees ? All modifications occur on a path. So it suffices to copy one path. This is the path copying method. 16 Example -- path copying ....... ........ 1 3 12 14 15 18 .. 20 21 28 40 16 17 Example -- path copying ....... ........ 1 3 12 14 15 18 . . 12 14 15 16 18 20 21 28 40 18 Path copying -- analysis Gives fully persistent search trees! O(log n) time for update and access O(log n) space per update Want the space bound to be proportional to the number of field modifications that the ephemeral update did. In case of search trees we want the space consumption of update to be O(1) (at least amortized). 19 Application -- planar point location Suppose that the Euclidian plane is subdivided into polygons by n line segments that intersect only at their endpoints. Given such polygonal subdivision and an on-line sequence of query points in the plane, the planar point location problem, is to determine for each query point the polygon containing it. Measure an algorithm by three parameters: 1) The preprocessing time. 2) The space required for the data structure. 3) The time per query. 20 Planar point location -- example 21 Planar point location -- example 22 Solving planar point location (Cont.) Partition the plane into vertical slabs by drawing a vertical line through each endpoint. Within each slab the lines are totally ordered. Allocate a search tree per slab containing the lines at the leaves with each line associate the polygon above it. Allocate another search tree on the x-coordinates of the vertical lines 23 Solving planar point location (Cont.) To answer query first find the appropriate slab Then search the slab to find the polygon 24 Planar point location -- example 25 Planar point location -- analysis Query time is O(log n) How about the space ? (n2) And so could be the preprocessing time 26 Planar point location -- bad example Total # lines O(n), and number of lines in each slab is O(n). 27 Planar point location & persistence So how do we improve the space bound ? Key observation: The lists of the lines in adjacent slabs are very similar. Create the search tree for the first slab. Then obtain the next one by deleting the lines that end at the corresponding vertex and adding the lines that start at that vertex How many insertions/deletions are there alltogether ? 2n 28 Planar point location & persistence (cont) Updates should be persistent since we need all search trees at the end. Partial persistence is enough Well, we already have the path copying method, lets use it. What do we get ? O(nlogn) space and O(nlog n) preprocessing time. We shall improve the space bound to O(n). 29 What are we after ? Break each operation into elementary access steps (ptr traversal) and update steps (assignments, allocations). Want a persistent simulation with consumes O(1) time per update or access step, and O(1) space per update step. 30 Making data structures persistent (DSST 89) We will show a general technique to make data structures partially and later fully persistent. The time penalty of the transformation would be O(1) per elementary access and update step. The space penalty of the transformation would be O(1) per update step. In particular, this would give us an O(n) space solution to the planar point location problem 31 The fat node method Every pointer field can store many values, each tagged with a version number. NULL 4 5 7 15 32 The fat node method (Cont.) Simulation of an update step when producing version i: • When a new node is created by the ephemeral update we create a new node, each value of a field in the new node is marked with version i. NULL 4 5 7 15 • When we change a value of a field f to v, we add an entry to the list of f with key i and value v 33 The fat node method (Cont.) Simulation of an access step when navigating in version i: • The relevant value is the one tagged with the largest version number smaller than i NULL 4 5 7 15 34 Partialy persistent deques via the fat node method 1 Null 1 x V1 Null Null 1 1 Null 2 x 2 y Null V2 = inject(y,V1) Null V3 = eject(V2) 2 Null 1 1 Null 2 x 2 y 2 3 Null Null 1 Null 1 2 x 4 2 y Null V4= inject(z,V3) 2 3 Null 4 z 4 Null 35 Fat node -- analysis Space is ok -- O(1) per update step That would give O(n) space for planar point location since each insertion/deletion does O(1) changes amortized. We screwed up the update time, it may take O(log m) to traverse a pointer, where m is the # of versions So query time goes up to O(log2n) and preprocessing time is O(nlog2n) 36 Node copying This is a general method to make pointer based data structures partially persistent. Nodes have to have bounded in degree and bounded outdegree We will show this method first for balanced search trees which is a slightly simpler case than the general case. Idea: It is similar to the fat node method just that we won’t make nodes too fat. 37 Partially persistent balanced search trees via node copying Here it suffices to allow one extra pointer field in each node Each extra pointer is tagged with a version number and a field name. When the ephemeral update allocates a new node you allocate a new node as well. When the ephemeral update changes a pointer field if the extra pointer is empty use it, otherwise copy the node. Try to store pointer to the new copy in its parent. If the extra ptr at the parent is occupied copy the parent and continue going up this way. 38 Insert into persistent 2-4 trees with node copying ....... ........ 1 3 12 14 18 .. 20 21 28 16 39 Insert into persistent 2-4 trees with node copying 1 ....... ........ 1 3 12 14 18 .. 20 21 28 12 14 16 18 40 Insert into persistent 2-4 trees with node copying 1 ....... ........ 1 3 12 14 18 .. 20 21 28 29 12 14 16 18 41 Insert into persistent 2-4 trees with node copying 1 ....... ........ 1 3 12 14 18 12 14 16 18 .. 20 21 28 20 21 28 29 42 Insert into persistent 2-4 trees with node copying 2 1 ....... ........ 1 3 12 14 18 12 14 16 18 .. 20 21 28 20 21 28 29 43 Node copying -- analysis The time slowdown per access step is O(1) since there is only a constant # of extra pointers per node. What about the space blowup ? O(1) (amortized) new nodes per update step due to nodes that would have been created by the ephemeral implementation as well. How about nodes that are created due to node copying when the extra pointer is full ? 44 Node copying -- analysis We’ll show that only O(1) of copings occur on the average per update step. Amorized space consumption = real space consumption + = #(used slots in live nodes) A node is live if it is reachable from the root of the most recent version. ==> Amortized space cost of node copying is 0. 45 Node copying in general Each persistent node has d + p + e + 1 pointers e = extra pointers p = predecessor pointers 1 = copy pointer. 4 5 6 7 11 live 46 Simulating an update step in node x When there is no free extra ptr in x copy x. When you copy node x, and x points to y, c(x) should point to y, update the corresponding predecessor ptr in y. Add x to the set S of copied nodes. (S contains no nodes initially) y x 7 7 11 c(x) 47 Node copying in general (cont) Take out a node x from S, go to nodes pointing to x and update then, maybe copying more nodes x 11 y 7 7 11 11 11 48 Node copying in general (cont) Take out a node x from S, go to nodes pointing to x and update then, maybe copying more nodes x 11 y 7 7 11 11 11 49 Node copying in general (cont) Take out a node x from S, go to nodes pointing to x and update then, maybe copying more nodes x 11 y 7 7 11 11 11 50 Node copying in general (cont) Take out a node x from S, go to nodes pointing to x and update then, maybe copying more nodes x 11 y 7 7 11 11 11 51 Node copying in general (cont) Remove any node x from S, for each node y indicated by a predecessor pointer in x find in y the live pointer to x. • If this ptr has version stamp i, replace it by a ptr to c(x). Update the corresponding reverse pointer • If this ptr has version stamp less than i, add to y a ptr to c(x) with version stamp i. If there is no room, copy y as before, and add it to S. Update the corresponding reverse pointer 52 Node copying (analysis) Actual space consumed is |S| = #(used extra fields in live nodes) = -e|S| + p|S| This is smaller than |S| if e > p (Actually e ≥ p suffices if we were more careful) So whether there were any copings or not the amortized space cost of a single update step is O(1) 53 The fat node method - full persistence Does it also work for full persistence ? NULL 1 5 5 6 7 6 We have a navigation problem. 54 The fat node method - full persistence (cont) Maintain a total order of the version tree. 5 6 7 8 9 5 7 6 8 5 7 9 6 8 55 The fat node method - full persistence (cont) When a new version is created add it to the list immediately after its parent. ==> The list is a preorder of the version tree. 56 The fat node method - full persistence (cont) When traversing a field in version i, the relevant value is the one recorded with a version preceding i in the list and closest to it. 5 NULL 1 6 5 7 8 9 6 5 7 6 5 7 9 8 6 8 57 The fat node method - full persistence (cont) How do we update ? 5 NULL 1 6 5 7 7 8 10 9 6 10 5 5 7 9 6 8 10 7 9 6 8 58 The fat node method - full persistence (cont) 5 NULL 1 6 7 10 5 7 8 9 6 10 5 5 7 9 6 8 10 7 9 6 8 So what is the algorithm in general ? 59 The fat node method - full persistence (cont) Suppose that when we create version i we change field f to have value v. Let i1 (i2) be the first version to the left (right) of i that has a value recorded at field f i1 i v f i2 i1 i i2 60 The fat node method - full persistence (cont) We add the pair (i,v) to the list of f Let i+ be the version following i in the version list v’ i1 i+ i v f i2 i1 i i+ i2 If (i+ < i2) or i+ exists and i2 does not exist add the pair (i+,v’) where v’ is the value associated with i1. 61 Fully persistent 2-4 trees with the fat node method 0 ....... ........ 1 3 12 14 18 .. 20 21 28 16 62 Insert into fully persistent 2-4 trees (fat nodes) 0 0 1 1 1 ....... ........ 1 3 12 14 18 .. 20 21 28 12 14 16 18 63 Insert into fully persistent 2-4 trees (fat nodes) 0 1 0 2 1 2 1 ....... ........ 1 3 12 14 18 .. 20 21 28 29 12 14 16 18 64 Insert into persistent 2-4 trees with node copying 0 1 0 2 1 2 2 1 ....... ........ 1 3 12 14 18 12 14 16 18 .. 20 21 28 20 21 28 29 65 Insert into persistent 2-4 trees with node copying 0 1 0 2 1 2 1 1 2 ....... ........ 1 3 12 14 18 12 14 16 18 .. 20 21 28 20 21 28 29 66 Fat node method (cont) How do we efficiently find the right value of a field in version i ? Store the values sorted by the order determined by the version list. Use a search tree to represent this sorted list. To carry out a find on such a search tree we need in each node to answer an order query on the version list. Use Dietz and Sleator’s data structure for the version list. 67 Fat node method (summary) We can find the value to traverse in O(log(m)) where m is the number of versions We get O(1) space increase per ephemeral update step O(log m) time slowdown per ephemeral access step 68 Node splitting Similar to node copying. (slightly more evolved) Allows to avoid the O(log m) time slowdown. Converts any pointer based data structure with constant indegrees and outdegrees to a fully persistent one. The time slowdown per access step is O(1) (amortized). The space blowup per update step is O(1) (amortized) 69 Search trees via node splitting You get fully persistent search trees in which each operation takes O(log n) amortized time and space. Why is the space O(log n) ? Since in the ephemeral settings the space consumption is O(1) only amortized. 70 Search trees via node splitting So what do we need in order to get persistent search trees with O(1) space cost per update (amortized) ? We need an ephemeral structure in which the space consumption per update is O(1) on the worst case. You can do it ! ==> Red-black trees with lazy recoloring 71 What about deques ? We can apply node splitting to get fully persistent deques with O(1) time per operation. We can also transform the simulation by stacks into a real time simulation and get O(1) time solution. What if we want to add the operation concatenate ? None of the methods seems to extend... 72