CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch

Discrete Algs for Mobile Wireless Sys Lecture 33 Topic: Data Aggregation in Sensor Networks Sources: Nath, Gibbons, Seshan & Anderson Shrivastava, Buragohain, Agrawal & Suri Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Aggregation Problem How to compute the answer to a query in a sensor network that requires aggregating data from all (or many) sensors? Example: Suppose the nodes take temperature readings and queries ask for min/max/average temperature Data has to flow through the network to the node that issues the query In some cases, data can be aggregated on the way save bandwidth and energy Example: to find max temp., each node propagates largest temp. it has learned about Discrete Algs for Mobile Wireless Sys

Communication to Support Aggregation Need to propagate sensor readings in some orderly way Example: send data over a spanning tree rooted at the querying node not robust: link or node failure will partition the tree, lose contact with sensors in subtree Prefer to use multipath routing (message is sent on several paths) redundancy provides more resilience But duplication causes problems for aggregation OK for max, but what about average? Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Overview of Algorithm Provides framework for synopses of the data to be sent over multiple paths and then reconstructing correct answer Phase 1: aggregate query is flooded through the network and an aggregation topology is constructed Phase 2: aggregate values are continually routed toward the querying node: each node converts its sensor data to a synopsis (SG function) nodes merge two synopses into one (SF function) querying node converts synopsis back to final answer (SE function) Discrete Algs for Mobile Wireless Sys

Specific Aggregation Topology Rings: kind of like levels in breadth-first search Nodes are partitioned into rings during Phase 1: querying node q is in ring 0 a node is in ring i if it receives the query first from a node in ring i–1 Phase 2 is divided into epochs, one aggregate answer per epoch each node in outer ring (farthest distance from q) computes s := SG(r), where r is its sensor reading, and broadcasts s each node in ring i computes s := SG(r), where r is its sensor reading, and updates s := SF(s,s'), where s' is each synopsis received from a neighbor, then broadcasts s querying node computes SE(s) Synchronous algorithm Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Figure R0 R1 R2 q A C B Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Analysis of Framework Complexity: each node broadcasts once per epoch Same as spanning-tree-based approach More resilient than spanning-tree-based approach Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys The Functions What should SG, SF, and SE be in order to give the "correct" answer? First, give a condition on the functions that is intuitive Then show there are 4 simple checks that can be done on proposed functions These conditions are necessary and sufficient to preserve correctness Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys ODI-Correctness Final result should be independent of how the data was routed to querier: same no matter in which order the readings are combined and how many times they are included (duplicated) during the routing Sensor reading r : <measurement, metadata> assumed to be unique Suppose we have SG, SF and SE Define synopsis label SL(s) = {r} if s = SG(r ) and SL(s) = SL(s1) Ums SL(s2) if s = SF(s1,s2) multiset union Discrete Algs for Mobile Wireless Sys

ODI-Correctness (cont'd) What constitutes a "duplicate" depends on what is being computed Ex: average temp vs. number of distinct temps q : multiset of sensor readings  set of (unique) values q(SL(s)) = set of unique values in all the sensor readings that formed the synopsis Discrete Algs for Mobile Wireless Sys

ODI-Correctness Definition Let {v1,…,vk} be set of values in the label of s, i.e., q(SL(s)). Then s must be same as computation on "canonical left-deep tree": s := SG(v1) for i = 2 to k do s := SF(s,SG(vi)) I.e., regardless of redundancy caused by multipath routing, the final synopsis is the same as if each distinct value is included just once Discrete Algs for Mobile Wireless Sys

ODI-Correctness Figure SG SF r1 r2 r3 r4 r5 s SG SF r1 r2 r3 r4 r5 s Aggregation DAG Canonical left-deep tree Discrete Algs for Mobile Wireless Sys

A Simple Test for ODI-Correctness duplicate preservation: q({r1}) = q({r2})  SG(r1) = SG(r2) if two readings are considered duplicates, then the same synopsis is generated commutativity: SF(s1,s2) = SF(s2,s1) associativity: SF(s1,SF(s2,s3)) = SF(SF(s1,s2),s3) idempotence: SF(s,s) = s Discrete Algs for Mobile Wireless Sys

More About the Conditions Theorem: The previous 4 conditions are necessary and sufficient for the SG and SF functions to ensure ODI-correctness. Proof Sketch: sufficiency: If SG and SF satisfy the 4 conditions, then show that any computation DAG can be transformed into a canonical left-deep binary tree that produces the same output necessity: Argue that the 4 conditions follow from the definition of ODI-correctness. Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Count Example Query: What is the (approximate) total number of sensor nodes in the network? Synopsis: a bit vector of length k > log N, where N is an upper bound on the number of nodes N could be original number of nodes deployed, or some function of the size of the id space Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys SG for Count Example No sensor is actually read for this example. Let SG return vector s[1..k], where a certain entry is 1 rest of the entries are 0 How to decide which entry should be 1: entry CT(k), where CT(k) is a random variable that returns value i with probability 1/2i, 1 ≤ i < k. How to compute CT(k): Toss a fair coin until either the first head occurs or k coin tosses have occurred with no heads; return number of tosses Give intuition for this definition of SG after seeing SF and SE Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Computation of CT(k) Why does the coin-tossing protocol give the desired random variable? Proof by Example: Suppose k = 4. First toss is H, and 1 is returned, with probability 1/2 Otherwise, second toss is H, and 2 is returned, with probability 1/4 Otherwise third toss is H and 3 is returned, with probability 1/8 (and then 4 is returned with probability 1/8, but the definition of CT(4) only cares about 1 through 3) Discrete Algs for Mobile Wireless Sys

SF and SE for Count Example SF(s,s'): s[i] := s[i] OR s'[i], 1 ≤ i ≤ k return s SE(s): return 2i-1/.77351, where i is the minimum index such that s[i] = 0 Discrete Algs for Mobile Wireless Sys

Intuition for Count Synopsis Functions Suppose all (live) sensors have a failure-free path to the querier. The final bit vector to which SE is applied indicates which bit positions have been set by at least one node The probability of n nodes failing to set the i-th bit is (1–2i)n by definition of SG Thus the number of (live) nodes is proportional to 2i–1 constant of proportionality is 1/.77351 Discrete Algs for Mobile Wireless Sys

Intuition for Count Synopsis Functions Alternatively… We expect half the nodes to set the 1st bit, a quarter of the nodes to set the 2nd bit, an eighth of the nodes to set the 3rd bit, etc. If there are n distinct nodes, then we might expect log n bits to be set I.e., if log n = i bits are set, then we might expect there to be about n = 2i nodes Discrete Algs for Mobile Wireless Sys

Count Algorithm is ODI-Correct Note that ODI-correctness says nothing about the SE function, only that SE will return the same result as in the canonical tree. "Clever algorithms are still required to get provably good approximations, although the task has been simplified…" Commutativity, associativity, and idempotence follow from properties of Boolean OR Discrete Algs for Mobile Wireless Sys

Count Algorithm is ODI-Correct Why does SG preserve duplicates? Assume each node calls SG only once. Show that if sensor readings are considered duplicates, then the synopsis generated by SG is the same. Since there is no actual sensor reading for this algorithm, we just use ids for the readings. Assumption that each node calls SG only once ensures the property. Discrete Algs for Mobile Wireless Sys

Implicit Acknowledgments When a node broadcasts a synopsis, avoid overhead of explicit acknowledgments from receivers this way: node u broadcasts its synopsis node u snoops (listens to) subsequent broadcasts by its parent nodes (nodes closer to the querying node) if the synopsis broadcast by a parent "effectively includes" u's synopsis, u does not need to rebroadcast, otherwise rebroadcast (or adapt the topology) Discrete Algs for Mobile Wireless Sys

Implicit Acknowledgments (cont'd) How can u accurately infer if its broadcasts was "effectively included"? Suppose u's synopsis was x and the parent's was z. If SF(x,z) = z, then x is effectively included. Why? Since SF is commutative, associative, and idempotent, it is a "semi-lattice". in a semi-lattice, every 2 elements x and y have a least upper bound z, and SF(x,z) = z = SF(y,z) Count example: check if appropriate bits are set Discrete Algs for Mobile Wireless Sys

Error Bounds of Approximate Answers Sources of error: communication error: some nodes have no failure-free propagation path to querier approximation error: introduced by SG, SF and SE functions. defined as relative error of computed answer w.r.t. exact algorithm using the same readings Argue that communication error can be made negligible by deploying sensor nodes sufficiently densely Discrete Algs for Mobile Wireless Sys

Error Bounds of Approximate Answers (cont'd) Approximation error analysis for the centralized data stream model work in this model, since synposis is ODI-correct canonical left-deep tree corresponds to processing a data stream of sensor readings in a centralized location Thus, e.g., Count algorithm has same approximation error guarantees as computed by Flajolet & Martin exact error for Count depends on number of independent bit vectors used per synopsis - see other papers for details Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys More Examples Max and Min: easy. SG is the value, SF takes larger/smaller, SE is identity Sum: cf. paper by Considine et al. which adapts Count algorithm Average, Standard deviation, Second Moment: cf. paper by Considine et al. which uses Sum Count Distinct: modification of Count Discrete Algs for Mobile Wireless Sys

Uniform Sample Example Compute a uniform sample of a given size K of the values occurring at all nodes in the network Synopsis: a sample of size K tuples (or fewer initially) SG: output (val,r,id) where val is the sensor reading of the node r is a random number drawn uniformly from [0,1] id is the node's id SF(s,s'): list the tuples in s U s' in decreasing order of r-value, and output the first K (or all, if less than K total) U is set union, removes duplicates SE(s): output the set of values in the tuples of s Discrete Algs for Mobile Wireless Sys

Uniform Sample Example (cont'd) SG labels each reading with a random number, thus placing it in a random position in the global ordering of all readings So taking first K in the ordering gives a uniform sample. Uniform sample can then be used… Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys More Examples Use uniform samples to compute these aggregates: k-th statistical moment (k = 1 is the mean) k-th percentile value (k = 50 is the median) with certain error and probability, by choosing the sample size appropriately (cf. Bar-Yossef et al.) Compute the k most frequent values (k = 1 is the mode): run an ODI-correct Count algorithm for each value Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Adapting the Topology If message loss is detected as occurring "too frequently", nodes can adapt the Ring topology Idea: use a heuristic that tries to assign a node u to a ring so that there are plenty of ndoes in the next ring to forward u's synopsis to the querier ODI-correct synopses are helpful: implicit acks are used to detect message loss energy-efficiently duplicates that occur during the adaptation of the topology are not a problem Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Simulation Results Extensive! Synopsis diffusion reduces answer errors in lossy environments helps address challenges from correlated node failures does not use significantly more power What topology to use? Adaptive Rings has same overhead as Rings but much better accuracy Adaptive Rings gets about 90% of the sensor readings most of the time vs. 100% with Flooding, but uses much less power Discrete Algs for Mobile Wireless Sys

Medians and Beyond [SBAS] Extend beyond min/max/sum the class of queries that can be answered in sensor networks to include approximate quantiles (including median) most frequent data values (including consensus) histogram of data distribution range queries Provide strict theoretical guarantees on the approximation quality of the answers in terms of message size Discrete Algs for Mobile Wireless Sys

Comparison with Nath Paper Some of the same problems are considered "Medians and Beyond" is concerned with efficiency of message size and its tradeoff with quality of approximation Nath paper was concerned with handling arbitrary ordering and duplicates "Medians and Beyond" assumes no duplicates Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Overview Assume we have a tree rooted at the querying node To compute Average: each node sends to its parent the sum of thedata values of its descendants and its number of descendants constant size messages To compute Median, need to keep track of all distinct values size of messages, and memory, grows linearly Trade off memory and bandwidth with accuracy of approximations Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Q-Digests Assume sensor readings are integers in the range [1,s] Introduce q-digest data structure to answer quantile queries with messages of size m error O((log s)/m) Users specify message size vs. error tradeoff q-digest measures maximum error accumulated so far Once q -digest query is done, use it to compute quantiles, data distribution,… Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys More on q-Digest Compute a compressed view of the complete distribution of values (instead of just a function of the values) Use this view of the distribution to compute approximations of various functions Basic idea: Essentially compute a histogram, but equally large, instead of equally spaced, buckets buckets can overlap size of buckets gives accuracy vs. communication tradeoff Discrete Algs for Mobile Wireless Sys

Definition of q-Digest Group values into variable-sized buckets of almost equal weights size refers to range weight refers to number of elements q-digest consists of a set of buckets Build a complete binary tree 1,…,s at the leaves every tree node is a bucket, its range is all the leaves in its subtree At any given point, only some of the buckets are being used Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Example 1 data range 1-8 15 data items 5 buckets 2 2 There are 4 3's, 6 4's, 2 values in {5,6}, 2 values in {7,8}, and 1 value between 1 and 8. 4 6 1 2 3 4 5 6 7 8 Discrete Algs for Mobile Wireless Sys

Definition of q-Digest Given compression parameter k and number of data items n, a (tree) node v is in the q-digest iff: count(v) ≤ n/k node should not have a high count count(v) + count(parent(v)) + count(sibling(v)) > n/k if a node and its children have low total count then combine using Compress algorithm For a leaf node, if count > n/k, then it is in the q-digest Root only needs to satisfy first condition Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Example 1 2 3 4 5 6 7 8 check that this has k = 5; n/k = 3 Each orange leaf node has at least 3 values; each orange non-leaf node has at most 3 values. Each orange non-root node, together with its sibling and parent, has more than 3 values. Each white node together with its parent and sibling has at most 3 values. Discrete Algs for Mobile Wireless Sys

Centralized Construction of q-Digest Go through all the tree nodes bottom up Check which ones satisfy the 2 properties. If a node v has a child that violates 2nd property then merge v with both its children Detailed info about values which occur frequently is preserved, while less frequently occurring values are lumped into larger buckets resulting in info loss Discrete Algs for Mobile Wireless Sys

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

Distributed Construction of q-Digest Represent a q-digest by numbering the nodes of the digest tree and sending a set of (node id, count) pairs q-digests move up the spanning tree, being merged as they go. To merge 2 q-digests: take their union add the counts of buckets with the same range compress the result Merging can cause information loss. Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Analysis of Q-Digest Lemma 1: A q-digest with parameter k has size (number of buckets) at most 3k. because the count of a node and its children can't be too small Lemma 2: In a q-digest with parameter k, the maximum error in the count of any node is n(log s)/k. because in the worst case the count of a node can deviate from the actual value by the sum of the counts of its ancestors Lemma 3: Merging multiple q-digests gives the same error as in Lemma 2. Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Quantile Queries Problem Statement: Given a fraction q between 0 and 1, find the value whose rank in sorted sequence of the n values is qn. Median is when q = 1/2 Relative error is defined to be |r – qn|/n, where r is the true rank of the returned value Discrete Algs for Mobile Wireless Sys

Using Q-Digest to Answer a Quantile Query Goal: find q-th quantile Sort the nodes of the q-digest in increasing order of max values (right endpoints); break ties by putting smaller ranges first this gives post-order traversal of the tree Scan sorted list and add up the counts Let v be the first node at which the running sum exceeds qn Return the max value of node v Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Error Analysis Answer returned is v.max There are at least qn values less than or equal to v.max, by choice of v Error comes from values that are less than v.max but are stored in ancestors of v (these buckets are listed after v) But this error is at most n(log s)/k Note that estimate is always at least as great as the eact answer Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Example 1 a a through o are the ids of the digest tree nodes: j = [3:3] k = [4:4] f = [5:6] g = [7:8] a = [1:8] b c 2 2 d e f g 4 6 h i j k l m n o 1 2 3 4 5 6 7 8 Find Median (q = 1/2); recall n = 15 so look for 7.5 Sorted list is (j,4), (k,6), (f,2), (g,2), (a,1) Running sums of counts are 4, 10 - done! Return max value in tree node k, which is 4 Error is at most sum of counts on path from k to root, which is 1 Discrete Algs for Mobile Wireless Sys

Trading Off Error and Message Size Memory and message size are controlled by the compression factor k: If k is small, then fewer buckets but wider range of values are lumped together If k is large, then more buckets but more fine-grained distribution of values to buckets If the maximum number of buckets you can afford is m, then set k = m/3 (by Lemma 1) and get error at most  = 3(log s)/m (by Lemmas 2 and 3) Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Other Queries Inverse Quantile: given a value x, determine its rank in the sorted sequence of input values Algorithm: construct same sorted list traverse list from beginning to end return as the answer the sum of the counts of buckets v for which x > v.max. Reported rank is between rank(x) and rank(x) + n Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Other Queries Range Query: find the number of values in the range [low,high]. Algorithm: perform inverse quantile queries to get the ranks of low and high return the difference in their ranks Maximum error is 2n Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Other Queries Consensus Query: Given a fraction f between 0 and 1, find all values that are reported by more than fn sensors Algorithm: Find all unit-width (leaf) buckets with count > (f–)n and return their values Since a leaf bucket's count has error at most n, this finds all values with frequency more than fn There may be some false positives: some values with count between (f–)n and fn may also be reported Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Confidence Factor Worst-case error is 3 (log s)/m, but it is unlikely that an execution will be this bad choosing message size m according to this constraint will be overkill and waste bandwidth Instead set m to a value for which it is expected that the error bound will be met Need to calculate the actual error in each q-digest: called confidence factor Define weight of a path: sum of counts of the nodes in the path Define confidence factor: maximum weight of any root-to-leaf path, divided by n Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Simulation Results Compared against simple scheme of keeping track of every distinct value together with its count q-digest scheme works well Discrete Algs for Mobile Wireless Sys

Discrete Algs for Mobile Wireless Sys Open Questions Continuous queries? Lost messages? Duplicate invariance? Include spatial information? Optimality of results? Discrete Algs for Mobile Wireless Sys