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Efficient ε-approximate summaries over a Sliding Window

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• Introduction
  • Arasu and Manku studied ε-approximate summaries over a data stream for two problems:

    Frequency count Quantile (ranking)

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  • They develop a "extension" technique that can maintain ε-approximate summaries by applying existing techniques multiple times

    Then they added a merge algorithm that combine the different layers into a coherent summary

    This summary will be an ε-approximate summaries

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• Definitions and Notations
  • Arasu and Manku has a different definition of a window

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  • Unbounded window: (no limit on the window size :-))

    An unbounded window sketch Sε is defined using the error parameter ε Sε allows ε-approximate statistics to be computed over a variable size window of the stream Sε(m, N) = content of the sketch Sε when: current size of the stream = m current size of the window = N

    Graphically:

  • Supported operations on S_ε(m, N):

    Insert: Constructs Sε(m+1, N+1) from Sε(m, N) by inserting the next element em+1 Delete: Constructs Sε(m, N-1) from Sε(m, N) Query: Returns an ε-approximate value (answer) over the bag of elements { em-N+1, em-N+2, ..., em}

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  • Bounded window: (window with a maximum size)

    An bounded window sketch SW,ε is defined using the error parameter ε and the maximum window size W SW,ε also allows ε-approximate statistics to be computed over a variable size window of the stream, but the window size is always ≤ W SW,ε(m, N) = content of the sketch SW,ε when: current size of the stream = m current size of the window = N

    Graphically:

  • Supported operations on S_ε(m, N):

    Insert: Constructs Sε(m+1, N+1) from Sε(m, N) by inserting the next element em+1 Delete: Constructs Sε(m, N-1) from Sε(m, N) Query: Returns an ε-approximate value (answer) over the bag of elements { em-N+1, em-N+2, ..., em}

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• My focus: fixed size window
  • Although the paper discusses variable sized windows, I will limit the discussion to only fixed sized windows - with constant width

  • It can easily be generalized to fixed sized windows - with width limitation

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• Important Assumption
  • The paper assumes that:

    W is some power of 2      (i.e., 2k)              1/ε is also some power of 2

  • The reason for these assumptions is that:

    the algorithm will maintain multiple summaries with different window sizes W that are half of each other...

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  • If W and 1/ε is not some power of 2; then use the following derived values:

    Use a value W' that is a power of 2 and: W ≤ W' ≤ 2×W Use the value 1/ε' such that: W' × ε' ≤ W× ε

  • Example:

    Suppose given: W = 1000 and ε = 0.01 Then use: W' = 1024 (Because 1000 (= W) ≤ 1024 ≤ 2000 (= 2W)) To find ε', solve: 1024 × ε' ≤ 1000 × 0.01 ==> ε' ≤ 10/1024 ==> ε' = 8/1024 ==> ε' = 2-7

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• Basic idea to Construct Frequency count/Quantile sketches over a FIXED SIZE window
  • Goal:: compute the ε-approximate count/quantile for a window of inputs

    input: a b c d e f g h i j k .... | | window: +---------------------+ (imagine that the window is huge) Window moves: input: a b c d e f g h i j k l m.... | | window: +---------------------+

    • We must do relatively little work to maintain ε-approximate statitics (count/quantile) for new window.

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  • Basic idea:

    Divide the input stream into blocks of equal sizes Current window +-----------------------------+ | | input: a b c d e f g h i j k l m .... | | | | | | | | +---------+ +---------+ +---------+ +---------+ block 1 block 2 block 3 block 4 Run Manku's algorithm or Greenwald's algorithm on each block to compute an ε'-approximate frequency count/ quantile NOTE: the accuracy that you need to run Manku or Greenwald on the values in each block is ε' and ε' will be different (smaller !) from ε !!! Design a method to combine the multiple summaries for different blocks to obtain an ε-approximate frequency count/quantile summary for the (entire) current window

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• Improved Idea...

  • Improved Idea

    So far, this idea is the same as the one in the previous webpage: click here This idea uses a one level block division and you need to use a lot of memory In order to keep the space needed for the summaries to a minimum, they use a multi-layered approach: They compute L+1 different summaries The summaries are categorized as layer 0, layer 1, ..., layer L Blocks in each layer is the same size (number of items in each block is same) Block size in the layer k + 1 is twice the block size in the layer k The block size in the last layer L is equal to the maximum window size W The number of layers L that are used is computed as follows: The block size in the last layer L is equal to the maximum window size W Since the block size of each lower layer is half that of the above layer, and the block size of layer 0 (N0) is equal to: N0 = W/(2L)              Because L can still be chosen, we can still decide the value of N0 !!! The size of the level 0 block (N0) can be chosen (set) to a specific value The authors chose N0 to be: N0 = (εW) / 4               NOTE: The factor "εW" in the above expression is necessary to ensure ε-approximate accuracy The factor εW is always a power of 2 (2q) !!! (εW)/4 is always an integer The factor "4" in the above expression is the smallest number that will make the method work. It is possible to use a larger constant other than 4 (e.g., 8, 16, ...) This will result in a smaller level 0 block size and the method will be less efficient So the best factor is 4 (gives the largest blocks that can provide the required accuracy) These 2 equations allow us to determine the number of layers L (+1): (εW/4)   =   W/(2L) or: 2L = 4/ε or: L = lg(4/ε) = 2 + lg(1/ε)

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  • Reason for a multi-layer approach:

    We use a different accuracy for each layer to take advantage of the different size of the blocks in different layers ! The size of the summary depends on: The accuracy ε' used in the layer The number of items in the blocks of the layer            The more items (larger) in a block, the more efficient is the summary data structure (It was use a relatively small number of entries to summarize a larger block)

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  • Keys to reducing memory requirement for storing summaries:

    Use smaller ε' values in larger blocks Because a summary is more efficient when blocks are large, the size of the summary will be relatively small Use larger ε' values in smaller blocks Because the summary is very inefficient when blocks are small, we use a relatively large ε' for force reduction of the size of the summary BUT: At the same time, we must prevent the (small number of) items in the block from destroying the ε-approximate guarantee of the summary !

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• Number of items in a layer l block
  • From the above discussion, we saw that the number of items in a layer 0 block is equal to:

    N0 = (εW) / 4

  • The number of items in a layer 1 block is twice as many, so:

    N1 = 21 × (εW)/4

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  • In general, the number of items in a layer k block is equal to:

    Nk = 2k × (εW) / 4

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• Error parameter for a layer l block
  • Important fact:

    Arasu's algorithm to construct summaries for a window will run the unchanged Manku's or Greenwald's algorithm L + 1 times - one algorithm for each layer BUT, using different error parameters

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  • The error-parameter ε_i set for a layer i block:

    2L 1 εi = ---- × -------- × ε 2i 4(L+1)

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    NOTES:

    εi decreases when i increases - the accuracy parameter for a higher layer block (with more elements in them), is smaller (more accurate). εi can be larger (!) than ε for small values if i !!! E.g.: if L=7, and i = 0, we have: εi = 27/(4×8) × ε = (128/32) × ε = 4 × ε

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• The error for data in the layer i block
  • Consider the expression ε_i × N_i which is the error guarantee of the summary for layer i

  • The expressions for ε_i (see click here) and for N_i (see click here) are:

    1 2L εi = ---- × ------ × ε 2i 4L+4 Ni = 2i × (εW)/4

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  • Therefore:

    εW 1 2L εi × Ni = 2i ----- × ---- -------- ε 4 2i 4(L+1) εW 2L = ---- × ------ ε (2L = 4/ε - see: click here) 4 4(L+1) εW 4/ε = ---- × ------ ε 4 4(L+1) εW εi × Ni = --------- 4L + 4

    NOTE:

    The error expression εW for the summary of a block at any layer is independent of the layer !!! (L is a constant !)

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• Some terminology in Arasu's Algorithm
  • Terminology used in block summary construction:

    A block is INACTIVE if no element in the block has arrived yet. - Obviously, INACTIVE blocks cannot be used to answer queries... A block is UNDER-CONSTRUCTION if some elements in the block resides inside the current window, but some element(s) of the block has not yet arrived - UNDER-CONSTRUCTION blocks are NOT USED to compute the query on a window W A block is ACTIVE if every element in the block reside inside the current window - only ACTIVE blocks are used to compute the answer for a query on a window W An active block becomes EXPIRED if at least one element of the block reside outside the current window

  • Example:

    The light-blue shaded blocks are all ACTIVE. The unshaded shaded blocks at left are EXPIRED The unshaded shaded blocks at right are either UNDER-CONSTRUCTION or INACTIVE

  • Example continued:

    One layer 0 ACTIVE block has become EXPIRED One layer 0 INACTIVE block has become UNDER-CONSTRUCTION

  • Example continued:

    More data arrive, but the state of the blocks did not changed...

  • Example continued:

    A layer 0 block has changed its state from UNDER-CONSTRUCTION to ACTIVE A layer 1 block has changed its state from UNDER-CONSTRUCTION to ACTIVE

  • Example continued:

    4 blocks (one from each layer) have changed from ACTIVE to EXPIRED !!

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• Merging the multi-layer sketches (summaries) to obtain an ε-approximate sketch (summary) for a window
  • The gist of Arasu's algorithm to compute an ε-approximate summary (frequency count or quantile) over a window is:

    A merge algorithm that combine the summaries in the various layers It must selects the best (i.e., in the higher layer) summaries stored in ACTIVE blocks that covers the current window.

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  • Example covering:

    Example of "Covering the current window" with the best ACTIVE blocks: always choose the blocks in the higher layer first

    NOTES:

    The best covering will select the ACTIVE blocks at the highest level first. Then selec the ACTIVE blocks at the next highest level that covers some uncovered portion of the window. Notice that not every element in the current window will be covered by the ACTIVE windows (See figure: elements in the left and right edges of the current window are not accounted for by any ACTIVE windows) These uncovered (unaccounted) elements will contribute to the total error Arasu's method will make certain that the error contributed by these uncovered elements will not be significant (that is done by making the blocks at the lowest layer sufficiently small...)

  • Arasu's method continues on the next webpage...