DOCSLIB.ORG

Multiresolution Priority Queues and Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Multiresolution Priority Queues and Applications Multiresolution Priority Queues and Applications Jordi Ros-Giralt, Alan Commike, Peter Cullen Richard Lethin {giralt, commike, cullen, lethin}@reservoir.com Reservoir Labs 632 Broadway Suite 803, New York, NY 10012 Abstract䴨 Priority queues are container data structures The work we present stems from research on essential to many high performance computing improving the performance of a priority queue by applications. In this paper, we introduce multiresolution introducing a trade-off to exploit an existing gap priority queues, a data structure that achieves greater between the applications’ requirements and the set of performance than the standard heap based implementation by trading off a controllable amount of priorities supported by a priority queue. With this resolution in the space of priorities. The new data objective, we introduce the concept of multiresolution structure can reduce the worst case performance of priority queue, a new container data structure that can inserting an element from O(log(n)) to O(log(r)) , where flexibly trade off a controllable amount of resolution in n is the number of elements in the queue and r is the the space of priorities in exchange for achieving number of resolution groups in the priority space. The greater performance. worst case cost of extracting the top element is O(1) . A simple example works as follows. Suppose that we When the number of elements in the table is high, the use a priority queue to hold jobs {job ,,...,job job } amortized cost to insert an element becomes O(1) . 1 2 n that need to be scheduled using a “shortest job first” 1. Iɴᴛʀᴏᴅᴜᴄᴛɪᴏɴ policy. Each job job takes a time to process equal to Priority queues are a type of container data structures i p , and the removal of an element from the queue designed to guarantee that the element at the front of i the queue is the greatest of all the elements it contains, returns the job with the smallest processing time . (Hence determines the priority of each according to some total ordering defined by their min{pi} pi priority. This allows for elements of higher priority to job.) A common approach is to use a binary heap based be extracted first, independently of the order in which implementation of a priority queue, which effectively they are inserted into the queue. Fast implementations provides support for all possible range of priorities pi of priority queues are important to many high in the space of real numbers . Assume now that the performance computing (HPC) applications and hence set of possible priority values pi is discretized into l optimizing their performance has been the subject of subsets and that the application only requires intense research. In graph searching, the algorithmic guaranteeing the ordering between elements belonging complexity of essential algorithms such as finding the to different subsets. A multiresolution priority queue is shortest path between two nodes in a graph rely a data structure that can provide better performance entirely on the performance of a priority queue (see for than a standard priority queue by supporting the instance Dijkstra’s algorithm [1], [2]). Other essential coarser grained resolution space consisting of l applications are found in the field of bandwidth subsets rather than the complete set of priorities. management. For instance, in the Internet, routers Multiresolution priority queues can also be understood often have insufficient capacity to transmit all packets as a generalization of standard priority queues in which at line rate and so they use priority queues in order to the total ordering requirement imposed on the set of assign higher preference to those packets that are most priorities is relaxed, allowing for elements in the queue critical in preserving the quality of experience (QoE) to be only partially ordered. Hence an alternative name of the network (e.g., packets carrying delay sensitive to refer to this type of container data structures would data such as voice). Priority queues are also important be partial order priority queues. in other areas of research including Huffman This paper is organized as follows. In Section 2, we compression codes, operating systems, Bayesian spam summarize some of the main historical results in the filtering, discrete optimization, simulation of colliding research area of high performance priority queues. particles, or artificial intelligence to name some Section 3 presents the body of our work, introducing a examples. formal definition of the concept of multiresolution This work was funded in part by the US Department of Energy under contract DE-SC0017184. 1 A patent application with Serial No. 62/512,438 has been submitted for this work. priority queues, providing pseudocode for the main entropy independent of the number of elements in the routines to insert and extract elements from the queue, queue, and this result leads to our O(1) bound. and analyzing the algorithm’s complexity. This section Our approach is perhaps closest to the data structures introduces also two variants of the base algorithm to known as bucket queues [7] and calendar queues [ 8 ]. support sliding priority windows for applications such When the set of priorities are integers and bounded to as event-driven simulators and a binary tree based C , a bucket queue can insert and extract elements in optimization applicable to general graph algorithms O(1) and O(C) , respectively. The resolution of this such as finding the shortest path between two nodes. data structure however cannot be tuned and it does not Section 4 describes our implementation of a support the general case of priorities in the set of real multiresolution priority queue and demonstrates how it numbers . Calendar queues areO(1) for both insert can help a real world HPC application achieve greater and remove operations, but to achieve these bounds performance than a traditional binary heap based they need to be balanced and they also do not support solution. We summarize the main conclusions of this changing the resolution of the priority set to optimize work in Section 5. the trade off performance versus accuracy to match the 2. Bᴀᴄᴋɢʀᴏᴜɴᴅ Wᴏʀᴋ application requirements. Priority queues have been the subject of intense Our data structure can also be understood as a type of research since J. W. J. Williams introduced the binary sketching data structure. These are data structures that heap data structure and the heapsort algorithm in 1964 can answer certain questions about the data they store [3]. Williams’ work lead to a very efficient extremely efficiently, at the price of an occasional implementation with a worst case cost to insert and error. Examples of sketching data structures are Bloom extract elements from the priority queue of O(log(n)) , filters [9], count-min sketch [10] and the LFN data where n is the number of elements in the queue. This structure [11] . is in many applications a preferred way to implement 3. Mᴜʟᴛɪ-Rᴇꜱᴏʟᴜᴛɪᴏɴ Pʀɪᴏʀɪᴛʏ Qᴜᴇᴜᴇꜱ priority queues due to its simplicity and good performance. Variations exist that can achieve better 3.1. Definition performance on some operations at the expense of A multiresolution priority queue (MR-PQ) is a adding more implementation complexity, such as container data structure that retrieves its elements Fibonacci heaps [4], which have an amortized cost of based on their priority and according to three inserting and removing an element of O(1) and parameters: a priority resolution pΔ and two priority O(log(n)) , respectively. limitspmin andpmax . The priority limits are chosen Soft heaps by Chazelle [5] relate to our work in that such that pmax > pmin and (pmax pmin) is divisible by they also achieve a constant amortized cost by pΔ . Unlike traditional priority queues, multiresolution introducing a certain amount of error. Specifically, a priority queues do not guarantee the total order of soft heap can break the information-theoretic barriers elements within a certain priority resolution p . of the problem by introducing error in the key space in Δ However, by trading off a small resolution, they can a way that the problem’s entropy is reduced. While soft achieve higher performance. In particular, because heaps provide the most asymptotically efficient known many real world applications do not require priority algorithm for finding minimum spanning trees [6], queues to support infinite resolution spaces, in many of their applications are limited in that they cannot these cases MR-PQs can be used to achieve greater guarantee an upper bound error on the number of performance at no cost, as we will show. elements in the queue. (Depending on the sequence of insertions and extractions, all keys could be corrupted.) More formally, an MR-PQ is a queue that at all times Instead, following on Chazelle’s analogy, a maintains the following invariant: multiresolution priority queue breaks the Property 1. Multiresolution Priority Queue (MR-PQ) information-theoretic barriers of the problem by Invariant. Let e and e be two arbitrary elements exploiting the multiresolution properties of its priority i j space. Since in many problems the entropy of the with prioritiespi andpj , respectively, where priority space is lower than the entropy of the key pmin ≤ pi < pmax and pmin ≤ pj < pmax . Then for all space, the end result is a container data structure with a possible states, a multiresolution priority queue ensures lower computational complexity. In particular, a that element ei is dequeued before element ej if the multi-resolutive priority space will have a constant following condition is true: 2 Bᴜɪʟᴅ(q) (1) 1 sentinel = alloc_element(); § 2 queue = sentinel; 3 queue->next = queue; Intuitively, an MR-PQ discretizes the priority space 4 queue->prev = queue; 5 qltsize = (pmax-pmin)/pdelta + 1; into a sequence of slots or resolution groups 6for i in [1, qltsize): 7 qlt[i] = NULL; [pmin,)pmin + pΔ , [pmin + pΔ,)pmin +2 pΔ , …, 8 qlt[0] = queue; [pmax pΔ,)pmax and prioritizes elements according to 9 queue->prio = pmin - pdelta; the slot in which they belong.
Load more

Recommended publications
  • An Alternative to Fibonacci Heaps with Worst Case Rather Than Amortized Time Bounds∗
    Relaxed Fibonacci heaps: An alternative to Fibonacci heaps with worst case rather than amortized time bounds¤ Chandrasekhar Boyapati C. Pandu Rangan Department of Computer Science and Engineering Indian Institute of Technology, Madras 600036, India Email: [email protected] November 1995 Abstract We present a new data structure called relaxed Fibonacci heaps for implementing priority queues on a RAM. Relaxed Fibonacci heaps support the operations find minimum, insert, decrease key and meld, each in O(1) worst case time and delete and delete min in O(log n) worst case time. Introduction The implementation of priority queues is a classical problem in data structures. Priority queues find applications in a variety of network problems like single source shortest paths, all pairs shortest paths, minimum spanning tree, weighted bipartite matching etc. [1] [2] [3] [4] [5] In the amortized sense, the best performance is achieved by the well known Fibonacci heaps. They support delete and delete min in amortized O(log n) time and find min, insert, decrease key and meld in amortized constant time. Fast meldable priority queues described in [1] achieve all the above time bounds in worst case rather than amortized time, except for the decrease key operation which takes O(log n) worst case time. On the other hand, relaxed heaps described in [2] achieve in the worst case all the time bounds of the Fi- bonacci heaps except for the meld operation, which takes O(log n) worst case ¤Please see Errata at the end of the paper. 1 time. The problem that was posed in [1] was to consider if it is possible to support both decrease key and meld simultaneously in constant worst case time.
    [Show full text]
  • Priority Queues and Binary Heaps Chapter 6.5
    Priority Queues and Binary Heaps Chapter 6.5 1 Some animals are more equal than others • A queue is a FIFO data structure • the first element in is the first element out • which of course means the last one in is the last one out • But sometimes we want to sort of have a queue but we want to order items according to some characteristic the item has. 107 - Trees 2 Priorities • We call the ordering characteristic the priority. • When we pull something from this “queue” we always get the element with the best priority (sometimes best means lowest). • It is really common in Operating Systems to use priority to schedule when something happens. e.g. • the most important process should run before a process which isn’t so important • data off disk should be retrieved for more important processes first 107 - Trees 3 Priority Queue • A priority queue always produces the element with the best priority when queried. • You can do this in many ways • keep the list sorted • or search the list for the minimum value (if like the textbook - and Unix actually - you take the smallest value to be the best) • You should be able to estimate the Big O values for implementations like this. e.g. O(n) for choosing the minimum value of an unsorted list. • There is a clever data structure which allows all operations on a priority queue to be done in O(log n). 107 - Trees 4 Binary Heap Actually binary min heap • Shape property - a complete binary tree - all levels except the last full.
    [Show full text]
  • Assignment 3: Kdtree ______Due June 4, 11:59 PM
    CS106L Handout #04 Spring 2014 May 15, 2014 Assignment 3: KDTree _________________________________________________________________________________________________________ Due June 4, 11:59 PM Over the past seven weeks, we've explored a wide array of STL container classes. You've seen the linear vector and deque, along with the associative map and set. One property common to all these containers is that they are exact. An element is either in a set or it isn't. A value either ap- pears at a particular position in a vector or it does not. For most applications, this is exactly what we want. However, in some cases we may be interested not in the question “is X in this container,” but rather “what value in the container is X most similar to?” Queries of this sort often arise in data mining, machine learning, and computational geometry. In this assignment, you will implement a special data structure called a kd-tree (short for “k-dimensional tree”) that efficiently supports this operation. At a high level, a kd-tree is a generalization of a binary search tree that stores points in k-dimen- sional space. That is, you could use a kd-tree to store a collection of points in the Cartesian plane, in three-dimensional space, etc. You could also use a kd-tree to store biometric data, for example, by representing the data as an ordered tuple, perhaps (height, weight, blood pressure, cholesterol). However, a kd-tree cannot be used to store collections of other data types, such as strings. Also note that while it's possible to build a kd-tree to hold data of any dimension, all of the data stored in a kd-tree must have the same dimension.
    [Show full text]
  • Rethinking Host Network Stack Architecture Using a Dataflow Modeling Approach
    DISS.ETH NO. 23474 Rethinking host network stack architecture using a dataflow modeling approach A thesis submitted to attain the degree of DOCTOR OF SCIENCES of ETH ZURICH (Dr. sc. ETH Zurich) presented by Pravin Shinde Master of Science, Vrije Universiteit, Amsterdam born on 15.10.1982 citizen of Republic of India accepted on the recommendation of Prof. Dr. Timothy Roscoe, examiner Prof. Dr. Gustavo Alonso, co-examiner Dr. Kornilios Kourtis, co-examiner Dr. Andrew Moore, co-examiner 2016 Abstract As the gap between the speed of networks and processor cores increases, the software alone will not be able to handle all incoming data without additional assistance from the hardware. The network interface controllers (NICs) evolve and add supporting features which could help the system increase its scalability with respect to incoming packets, provide Quality of Service (QoS) guarantees and reduce the CPU load. However, modern operating systems are ill suited to both efficiently exploit and effectively manage the hardware resources of state-of-the-art NICs. The main problem is the layered architecture of the network stack and the rigid interfaces. This dissertation argues that in order to effectively use the diverse and complex NIC hardware features, we need (i) a hardware agnostic representation of the packet processing capabilities of the NICs, and (ii) a flexible interface to share this information with different layers of the network stack. This work presents the Dataflow graph based model to capture both the hardware capabilities for packet processing of the NIC and the state of the network stack, in order to enable automated reasoning about the NIC features in a hardware-agnostic way.
    [Show full text]
  • Priorityqueue
    CSE 373 Java Collection Framework, Part 2: Priority Queue, Map slides created by Marty Stepp http://www.cs.washington.edu/373/ © University of Washington, all rights reserved. 1 Priority queue ADT • priority queue : a collection of ordered elements that provides fast access to the minimum (or maximum) element usually implemented using a tree structure called a heap • priority queue operations: add adds in order; O(log N) worst peek returns minimum value; O(1) always remove removes/returns minimum value; O(log N) worst isEmpty , clear , size , iterator O(1) always 2 Java's PriorityQueue class public class PriorityQueue< E> implements Queue< E> Method/Constructor Description Runtime PriorityQueue< E>() constructs new empty queue O(1) add( E value) adds value in sorted order O(log N ) clear() removes all elements O(1) iterator() returns iterator over elements O(1) peek() returns minimum element O(1) remove() removes/returns min element O(log N ) Queue<String> pq = new PriorityQueue <String>(); pq.add("Stuart"); pq.add("Marty"); ... 3 Priority queue ordering • For a priority queue to work, elements must have an ordering in Java, this means implementing the Comparable interface • many existing types (Integer, String, etc.) already implement this • if you store objects of your own types in a PQ, you must implement it TreeSet and TreeMap also require Comparable types public class Foo implements Comparable<Foo> { … public int compareTo(Foo other) { // Return > 0 if this object is > other // Return < 0 if this object is < other // Return 0 if this object == other } } 4 The Map ADT • map : Holds a set of unique keys and a collection of values , where each key is associated with one value.
    [Show full text]
  • Programmatic Testing of the Standard Template Library Containers
    Programmatic Testing of the Standard Template Library Containers y z Jason McDonald Daniel Ho man Paul Stro op er May 11, 1998 Abstract In 1968, McIlroy prop osed a software industry based on reusable comp onents, serv- ing roughly the same role that chips do in the hardware industry. After 30 years, McIlroy's vision is b ecoming a reality. In particular, the C++ Standard Template Library STL is an ANSI standard and is b eing shipp ed with C++ compilers. While considerable attention has b een given to techniques for developing comp onents, little is known ab out testing these comp onents. This pap er describ es an STL conformance test suite currently under development. Test suites for all of the STL containers have b een written, demonstrating the feasi- bility of thorough and highly automated testing of industrial comp onent libraries. We describ e a ordable test suites that provide go o d co de and b oundary value coverage, including the thousands of cases that naturally o ccur from combinations of b oundary values. We showhowtwo simple oracles can provide fully automated output checking for all the containers. We re ne the traditional categories of black-b ox and white-b ox testing to sp eci cation-based, implementation-based and implementation-dep endent testing, and showhow these three categories highlight the key cost/thoroughness trade- o s. 1 Intro duction Our testing fo cuses on container classes |those providing sets, queues, trees, etc.|rather than on graphical user interface classes. Our approach is based on programmatic testing where the number of inputs is typically very large and b oth the input generation and output checking are under program control.
    [Show full text]
  • Readings Findmin Problem Priority Queue
    Readings • Chapter 6 Priority Queues & Binary Heaps › Section 6.1-6.4 CSE 373 Data Structures Winter 2007 Binary Heaps 2 FindMin Problem Priority Queue ADT • Quickly find the smallest (or highest priority) • Priority Queue can efficiently do: item in a set ›FindMin() • Applications: • Returns minimum value but does not delete it › Operating system needs to schedule jobs according to priority instead of FIFO › DeleteMin( ) › Event simulation (bank customers arriving and • Returns minimum value and deletes it departing, ordered according to when the event › Insert (k) happened) • In GT Insert (k,x) where k is the key and x the value. In › Find student with highest grade, employee with all algorithms the important part is the key, a highest salary etc. “comparable” item. We’ll skip the value. › Find “most important” customer waiting in line › size() and isEmpty() Binary Heaps 3 Binary Heaps 4 List implementation of a Priority BST implementation of a Priority Queue Queue • What if we use unsorted lists: • Worst case (degenerate tree) › FindMin and DeleteMin are O(n) › FindMin, DeleteMin and Insert (k) are all O(n) • In fact you have to go through the whole list • Best case (completely balanced BST) › Insert(k) is O(1) › FindMin, DeleteMin and Insert (k) are all O(logn) • What if we used sorted lists • Balanced BSTs › FindMin and DeleteMin are O(1) › FindMin, DeleteMin and Insert (k) are all O(logn) • Be careful if we want both Min and Max (circular array or doubly linked list) › Insert(k) is O(n) Binary Heaps 5 Binary Heaps 6 1 Better than a speeding BST Binary Heaps • A binary heap is a binary tree (NOT a BST) that • Can we do better than Balanced Binary is: Search Trees? › Complete: the tree is completely filled except • Very limited requirements: Insert, possibly the bottom level, which is filled from left to FindMin, DeleteMin.
    [Show full text]
  • Chapter C4: Heaps and Binomial / Fibonacci Heaps
    Chapter C4: Heaps and Binomial / Fibonacci Heaps C4.1 Heaps A heap is the standard implementation of a priority queue. A min-heap stores values in nodes of a tree such that heap-order property: for each node, its value is smaller than or equal to its children’s So the minimum is on top. (A max-heap can be defined similarly.) The standard heap uses a binary tree. Specifically, it uses a complete binary tree, which we define here to be a binary tree where each level except the last is full, and in the last level nodes are added left to right. Here is an example: 7 24 19 25 56 68 40 29 31 58 Now, any binary tree can be stored in an array using level numbering. The root is in cell 0, cells 1 and cells 2 are for its children, cells 3 through 6 are for its grandchildren, and so on. Note that this means a nice formula: if a node is in cell x,thenitschildren are in 2x+1 and 2x+2. Such storage can be (very) wasteful for a general binary tree; but the definition of complete binary tree means the standard heap is stored in consecutive cells of the array. C4.2 Heap Operations The idea for insertion is to: Add as last leaf, then bubble up value until heap-order property re-established. Insert(v) add v as next leaf while v<parent(v) { swapElements(v,parent(v)) v=parent(v) } One uses a “hole” to reduce data movement. Here is an example of inserting the value 12: 7 7 24 19 12 19 ) 25 56 68 40 25 24 68 40 29 31 58 12 29 31 58 56 The idea for extractMin is to: Replace with value from last leaf, delete last leaf, and bubble down value until heap-order property re-established.
    [Show full text]
  • Nearest Neighbor Searching and Priority Queues
    Nearest neighbor searching and priority queues Nearest Neighbor Search • Given: a set P of n points in Rd • Goal: a data structure, which given a query point q, finds the nearest neighbor p of q in P or the k nearest neighbors p q Variants of nearest neighbor • Near neighbor (range search): find one/all points in P within distance r from q • Spatial join: given two sets P,Q, find all pairs p in P, q in Q, such that p is within distance r from q • Approximate near neighbor: find one/all points p’ in P, whose distance to q is at most (1+ε) times the distance from q to its nearest neighbor Solutions Depends on the value of d: • low d: graphics, GIS, etc. • high d: – similarity search in databases (text, images etc) – finding pairs of similar objects (e.g., copyright violation detection) Nearest neighbor search in documents • How could we represent documents so that we can define a reasonable distance between two documents? • Vector of word frequency occurrences – Probably want to get rid of useless words that occur in all documents – Probably need to worry about synonyms and other details from language – But basically, we get a VERY long vector • And maybe we ignore the frequencies and just idenEfy with a “1” the words that occur in some document. Nearest neighbor search in documents • One reasonable measure of distance between two documents is just a count of words they share – this is just the point wise product of the two vectors when we ignore counts.
    [Show full text]
  • Rethinking Host Network Stack Architecture Using a Dataflow Modeling Approach
    Research Collection Doctoral Thesis Rethinking host network stack architecture using a dataflow modeling approach Author(s): Shinde, Pravin Publication Date: 2016 Permanent Link: https://doi.org/10.3929/ethz-a-010657596 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library DISS.ETH NO. 23474 Rethinking host network stack architecture using a dataflow modeling approach A thesis submitted to attain the degree of DOCTOR OF SCIENCES of ETH ZURICH (Dr. sc. ETH Zurich) presented by Pravin Shinde Master of Science, Vrije Universiteit, Amsterdam born on 15.10.1982 citizen of Republic of India accepted on the recommendation of Prof. Dr. Timothy Roscoe, examiner Prof. Dr. Gustavo Alonso, co-examiner Dr. Kornilios Kourtis, co-examiner Dr. Andrew Moore, co-examiner 2016 Abstract As the gap between the speed of networks and processor cores increases, the software alone will not be able to handle all incoming data without additional assistance from the hardware. The network interface controllers (NICs) evolve and add supporting features which could help the system increase its scalability with respect to incoming packets, provide Quality of Service (QoS) guarantees and reduce the CPU load. However, modern operating systems are ill suited to both efficiently exploit and effectively manage the hardware resources of state-of-the-art NICs. The main problem is the layered architecture of the network stack and the rigid interfaces. This dissertation argues that in order to effectively use the diverse and complex NIC hardware features, we need (i) a hardware agnostic representation of the packet processing capabilities of the NICs, and (ii) a flexible interface to share this information with different layers of the network stack.
    [Show full text]
  • Jt-Polys-Cours-11.Pdf
    Notes de cours Standard Template Library ' $ ' $ Ecole Nationale d’Ing´enieurs de Brest Table des mati`eres L’approche STL ................................... 3 Les it´erateurs .................................... 15 Les classes de fonctions .......................... 42 Programmation par objets Les conteneurs ................................... 66 Le langage C++ Les adaptateurs ................................. 134 — Standard Template Library — Les algorithmes g´en´eriques ...................... 145 Index ........................................... 316 J. Tisseau R´ef´erences ...................................... 342 – 1996/1997 – enib c jt ........ 1/344 enib c jt ........ 2/344 & % & % Standard Template Library L’approche STL ' $ ' $ L’approche STL Biblioth`eque de composants C++ g´en´eriques L’approche STL Fonctions : algorithmes g´en´eriques 1. Une biblioth`eque de composants C++ g´en´eriques sort, binary search, reverse, for each, accumulate,... 2. Du particulier au g´en´erique Conteneurs : collections homog`enes d’objets 3. Des indices aux it´erateurs, via les pointeurs vector, list, set, map, multimap,... 4. De la g´en´ericit´edes donn´ees `ala g´en´ericit´edes structures It´erateurs : sorte de pointeurs pour inspecter un conteneur input iterator, random access iterator, ostream iterator,... Objets fonction : encapsulation de fonctions dans des objets plus, times, less equal, logical or,... Adaptateurs : modificateurs d’interfaces de composants stack, queue, priority queue,... enib c jt ........ 3/344 enib c jt ........ 4/344 & % & % L’approche STL L’approche STL ' $ ' $ Du particulier . au g´en´erique int template <class T> max(int x, int y) { return x < y? y : x; } const T& max(const T& x, const T& y) { return x < y? y : x; } int template <class T, class Compare> max(int x, int y, int (*compare)(int,int)) { const T& return compare(x,y)? y : x; max(const T& x, const T& y, Compare compare) { } return compare(x, y)? y : x; } enib c jt .......
    [Show full text]
  • IBM Spectrum Scale 5.1.0: Concepts, Planning, and Installation Guide Summary of Changes
    IBM Spectrum Scale Version 5.1.0 Concepts, Planning, and Installation Guide IBM SC28-3161-02 Note Before using this information and the product it supports, read the information in “Notices” on page 551. This edition applies to version 5 release 1 modification 0 of the following products, and to all subsequent releases and modifications until otherwise indicated in new editions: • IBM Spectrum Scale Data Management Edition ordered through Passport Advantage® (product number 5737-F34) • IBM Spectrum Scale Data Access Edition ordered through Passport Advantage (product number 5737-I39) • IBM Spectrum Scale Erasure Code Edition ordered through Passport Advantage (product number 5737-J34) • IBM Spectrum Scale Data Management Edition ordered through AAS (product numbers 5641-DM1, DM3, DM5) • IBM Spectrum Scale Data Access Edition ordered through AAS (product numbers 5641-DA1, DA3, DA5) • IBM Spectrum Scale Data Management Edition for IBM® ESS (product number 5765-DME) • IBM Spectrum Scale Data Access Edition for IBM ESS (product number 5765-DAE) Significant changes or additions to the text and illustrations are indicated by a vertical line (|) to the left of the change. IBM welcomes your comments; see the topic “How to send your comments” on page xxxii. When you send information to IBM, you grant IBM a nonexclusive right to use or distribute the information in any way it believes appropriate without incurring any obligation to you. © Copyright International Business Machines Corporation 2015, 2021. US Government Users Restricted Rights – Use,
    [Show full text]