The Geometry of Browsing
Abstract
We present a geometric counting problem that arises in browsing and
solve it in constant time per query using nonexhaustive tables. On
the other hand, we prove that several closely related problems require
exhaustive tables, no matter how much time we allow per query.
1. Introduction
In this paper we address some algorithmic problems that arise in
connection with browsing a really large collection of datasets. The
interface paradigms we present have been developed and usertested as
part of the visual data mining effort (cf. [S]) at
the HumanComputer Interaction Laboratory (HCIL) in the University of
Maryland. The target application is the EOSDIS collection of datasets
in development by the U.S. National Aeronautics and Space
Administration (NASA).
What is EOSDIS?
The NASA EOSDIS (Earth Observing System Data and Information System)
project is attempting to provide online access to a rapidly growing
archive of scientific data about the Earth's land, water, and
air. Data is collected from satellites, ships, aircraft, and ground
crews, and stored in designated archive centers. Scientists,
teachers, and the general public are given access to this data via the
Internet.
HCIL is developing user interfaces that use the ``dynamic query''
paradigm and the ``query preview'' paradigm to facilitate the browsing
and retrieval of data from this very large archive. The background
for interfaces to EOSDIS is discussed further on the web at [H95].
What are Dynamic Queries?
Dynamic queries are an interface paradigm that allow the user to
interactively control query parameters and generate a rapidly animated
visual display of database search
results [AS94, AW95, E94, FS95, GR94, H, I96, I, S94, TBS96, WS92].
As users adjust sliders or buttons, results are updated nearly
continuously on the display. Each adjustment to a slider and each
button click is called a query. The answer to the query is
presented graphically. Experimental results have shown that dynamic
queries are a fast, effective, fun, and easytouse tool for novice
and expert users to find trends and spot exceptions [AW95, WS92].
Dynamic query user interfaces apply the principles of direct
manipulation to query formulation and provide:
 a visual representation of the query and the results,
 rapid, incremental, and reversible actions,
 selection by pointing (not typing), and
 immediate and continuous display of results.
Some demos of dynamic queries are available from [H, I].
What are Query Previews?
In a networked information system, there are three major obstacles
facing users in a querying process: slow network performance, large
data volume, and data complexity. EOSDIS has all of these. The
collection is predicted to reach into the petabytes (10^{15} bytes).
Additionally, EOSDIS datasets have numerous attributes, such as when
and where it was collected, and the types of features or measurements
in the dataset. With a formsbased interface, finding a particular
dataset or a group of datasets that match certain characteristics
would typically involve several iterations of querying and waiting for
results over the network. This would not only slow down the process of
finding datasets, but would slow down the network and servers for all
users.
Query previews, as developed in [DPS96], take advantage of
the fact that in many cases, perhaps most, users are only interested
in a small subset of the entire collection. For example, a user might
only be interested in data for Europe, instead of the whole world.
Narrowing the scope of the collection can greatly improve the
efficiency of browsing and querying. A query previewer gives the user
overviews of the entire collection such as a map showing the
distribution of datasets over the Earth. The total number of datasets
is also displayed. Using a dynamicquery interface, the user can
narrow his search to a selected region of latitude and longitude by
adjusting sliders. With another slider, the user can narrow his
search to a certain range of years. With checkboxes, the user can
narrow his search to only those datasets containing selected
attributes (for example, temperature and pressure). As search
parameters are adjusted (query), the distribution of datasets and
the total count are updated.
Once the scope of the search is sufficiently narrow, i.e., the number
of datasets matching the dynamic query is manageably small, the user
and the system are ready for more detailed query and exploration.
This second query phase, called ``query refinement, '' is another
dynamicquery interface to a more detailed view of the datasets
selected by the query preview. The details of query refinement are
independent of the query previewer and will not be addressed in this
paper.
Practical Requirements.
Queries will be answered via a computation that consults a
table of summary information about all datasets. Since the query
previewer is a dynamicquery interface, updates should ideally appear
to be continuous. Studies suggest that most users will in fact
tolerate a delay of at most 0.1 or 0.2 seconds per
update [AS94]. Since even this is much slower than typical
worldwideweb turnaround, it is necessary that the aforementioned
table be stored on the user's own node. Thus, diskspace limitations
and download times both dictate that the table not be very large; 1
megabyte seems like a good rule of thumb. To summarize, we need small
tables that support fast querying.
2. The QueryPreviewing Problem
In the EOSDIS example, each dataset is described by a 4dimensional
record whose fields indicate the scope of information that the dataset
contains: (1) range of latitude, (2) range of longitude, (3) range of
years, and (4) set of attributes. The user's query is also a record of
this type. An EOSDIS dataset matches the record if its range of
latitudes overlaps with the user's range of latitudes, and so on for
the other three dimensions. The result of the query is the number of
EOSDIS datasets that match the query.
In order to compute the distribution of datasets over the Earth, the
map is partitioned into squares and one query of the type described
above is evaluated for each square in order to determine the number of
datasets with information about that region of the Earth. Because we
will, in fact, give a constanttime querying algorithm, the partition of
the Earth into squares can in practice be very fine.
Geometric Interpretation
The first three fields in a record are ranges of numbers, so they can
be represented as intervals. Let us ignore temporarily the fourth
field (set of attributes). Then the record can be represented as a
3dimensional rectangle, the Cartesian (cross) product of those three
intervals. An EOSDIS record matches the query if the two
corresponding rectangles overlap, i.e, if their intersection is
nonempty. Please forgive us for belaboring an obvious point:
rectangle overlap is a logical AND, i.e., two rectangles overlap if
the intervals along the first dimension overlap and the intervals
along the second dimension overlap and the intervals along the third
dimension overlap.
Let us return to the fourth field in our EOSDIS records, the set of
attributes. We will represent each attribute by a number, so the
fourth field is a set of numbers. At the risk of forcing the geometric
metaphor, we will call a set of numbers a generalized interval.
Because the Earth is round, it may also be necessary to consider
intervals in dimension (2) that wrap around from the right edge of the
map to the left edge. These are called wrapped intervals.
Formal Problem Statements
In general we will consider records with d fields, giving rise to
ddimensional problems.
Definition 1:

N = {0, 1, . . . }, the set of natural numbers

N^{d} is the set of all ddimensional lattice
points in the 1st quadrant

An interval is a set {a, a+1, . . . , b} of
consecutive natural numbers.

A wrapped interval in {1, . . . , m} is either an interval or
the union of two intervals that contain 1 and m.

A generalized interval is a subset of N.

A rectangle in N^{d} is a crossproduct I_{1} x . . . x I_{d}
of d intervals I_{1}, . . . , I_{d}.

A wrapped rectangle in N^{d} is a crossproduct
I_{1} x . . . x I_{d} of d
wrapped intervals I_{1}, . . . , I_{d}.

A generalized rectangle in N^{d} is a crossproduct
I_{1} x . . . x I_{d} of d
generalized intervals I_{1}, . . . , I_{d}.

A intersects B if the intersection of A
and b is nonempty.

A is skew to B if there is no hyperplane parallel to
the coordinate axes that intersects both A and B.
In this paper, we will be mainly interested in three problems:
Rectangle Intersection (logical AND)
 Data to be preprocessed: A list D of rectangles in N^{d}
 Problem instance: A single rectangle Q in N^{d}
 Question:How many elements of D intersect Q?
Wrapped Rectangle Intersection (logical AND)
 Data to be preprocessed: A list D of wrapped rectangles in {1, . . . , m}^{d}
 Problem instance: A single wrapped rectangle Q in {1, . . . , m}^{d}
 Question: How many elements of D intersect Q?
Generalized Rectangle Intersection (logical AND)
 Data to be preprocessed: A list D of generalized rectangles in N^{d}
 Problem instance: A single generalized rectangle Q in N^{d}
 Question: How many elements of D intersect Q?
Although we do not state it as a formal problem, in practice we are
interested in the mixed case, where some dimensions of the records are
intervals, others are wrapped intervals, and yet others are
generalized intervals. Our results for the homogeneous problems
described above are directly applicable to the mixed case.
The following problem corresponds to queries based on logical OR
rather than logical AND. Because such queries are sometimes useful in
browsing, we consider them as well.
Rectangle Nonskewness (logical OR)
 Data to be preprocessed: A list D of rectangles in N^{d}
 Problem instance: A single rectangle Q in N^{d}
 Question: How many elements of D are not skew to Q?
Complexity Bounds
Throughout, let R denote a fixed rectangle in N^{d} that contains
each element of D. We present algorithms for rectangle intersection
and rectangle nonskewness that use tables whose size depends only on
the dimension d and the bounding rectangle R, and answer queries
in time that depends only on d. The preprocessing time depends on
D, but the cost for adding a single dataset to the list depends only
on d and R.
Results.
Assume that R is an n_{1} x . . . x n_{d} rectangle. Let
ρ = (2n_{1}  1) . . . (2n_{d}  1) < 2^{d} R.
 Rectangle Intersection can be solved with O(ρ) preprocessing
per element of D, using tables of size ρ, in time
O(d2^{d}) per query.
 Rectangle Nonskewness can be solved with O(ρ) preprocessing
per element of D, using tables of size ρ, in time
O(d4^{d}) per query.
 Generalized Rectangle Intersection requires exhaustive
tables (size 2^{n1}2^{n2} . . . 2^{nd}).
 Wrapped Interval Intersection requires exhaustive tables
(size n_{1}(n_{1}  1)n_{2}(n_{2}  1) . . . n_{d}(n_{d}  1)).
3. Geometric Algorithms
We identify each rectangle in N^{d} with the polytope obtained upon
replacing each of its points p with a unit dcube centered at p.
A face of a boundedpolytope S is interior to S if it is not
the ``exterior face'' and it is not entirely contained in the boundary
of S. Let F_{k}(S) denote the number of kdimensional faces of
S and let F*_{k}(S) denote the number of kdimensional faces
interior to S.
Lemma:
If S is a bounded, connected ddimensional polytope then
Σ_{0 ≤ k ≤ d}(1)^{dk} F*_{k}(S)} = 1.
Proof:
Euler's theorem (see, for example, [H69]) states that
Σ_{0 ≤ k ≤ d}(1)^{k} F_{k}(S) = 1 + (1)^{d}.
Let B denote the boundary of S. Then B is a connected
(d  1)dimensional polytope, so by Euler's theorem
Σ_{0 ≤ k ≤ d  1}(1)^{k} F_{k}(B)} = 1 + (1)^{d  1}.
Therefore
Σ_{0 ≤ k ≤ d}(1)^{k} F*_{k}(S)
 =  Σ_{0 ≤ k ≤ d}(1)^{k} F_{k}(S)  (1)^{d}  &Sigma_{0 ≤ k ≤ d  1}}(1)^{k} F_{k}(B)

 =  1 + (1)^{d}  (1)^{d}  (1 + (1)^{d  1})

 =  (1)^{d  1}

 =  (1)^{d}

so Σ_{0 ≤ k ≤ d}(1)^{dk} F*_{k}(S) = (1)^{d} (1)^{d} = 1.
For each 0, 1, . . . , or ddimensional cube c,
let #(c) denote the number of rectangles r in the list D such that
the interior of r intersects the interior of c. In particular,
the answer to the query Q is #(q).
We have
#(q) = Σ_{0 ≤ k ≤ d} (1)^{dk} Σ_{c is a kdimensional unit cube in the interior of Q} #(c)
 (1)

Why? Because each rectangle in D that does not intersect Q
contributes 0 to the sum, and each rectangle in D that intersects
Q contributes 1 to the sum.
Let dim(r) denote the dimension of a rectangle. If we stored
(1)^{ddim(c)}#(c) for each unit cube c, then we could
compute the sum specified in Equation (1) by summing over
each unit cube contained in Q. Better yet, as noted
in [B92], if we store ddimensional prefix sums, we
can evaluate that sum in constant time. For each unit cube a we
store Σ_{b ≤ a}(1)^{ddim(b)}#(b), where the
inequality must hold on every coordinate. Given such a table, the sum
specified in Equation (1) may be obtained with 2^{d}  2
additions and subtractions, by the principle of inclusion and
exclusion. The total storage needed is the number of unit cubes
interior to R whose dimension is d or less, which is exactly
ρ.
For concreteness we present the table construction and the query
algorithm for the case d=2 below. In order to simplify the
algorithm, we have used a table of size
2^{d}R, which is larger than we claimed. This
allows us to store 0s along the edges and avoid special cases.
procedure CountRecord(x_{1}, x_{2}, y_{1}, y_{2})
for i = 2x_{1}  1 to 2x_{2}  1 do
for j = 2y_{1}  1 to 2y_{2}  1 do
table[i, j] = table[i, j] + (1)^{i+1}(1)^{j+1}
end
procedure CountAllRecords
for i = 0 to 2n_{1}  1 do
for j = 0 to 2n_{2}  1 do
table[i, j] = 0
for each rectangle (x_{1}, x_{2}, y_{1}, y_{2}) in the list D do
CountRecord(x_{1}, x_{2}, y_{1}, y_{2})
end
procedure ComputePartialSums
for i = 2 to 2n_{1}  1 do
for j = 1 to 2n_{2}  1 do
table[i, j] = table[i  1, j] + table[i, j]
for i = 1 to 2n_{1}  1 do
for j = 2 to 2n_{2}  1 do
table[i, j] = table[i, j  1] + table[i, j]
end
procedure BuildTable
CountAllRecords
ComputePartialSums
end
function IncludeExclude(a_{1}, a_{2}, b_{1}, b_{2})
return table(a_{2}, b_{2})  table(a_{2}, b_{1})  table(a_{1}, b_{2}) + table(a_{1}, b_{1})
end
function Query(x_{1}, x_{2}, y_{1}, y_{2})
return IncludeExclude(2x_{1}  2, 2x_{2}  1, 2y_{1}  2, 2y_{2}  1)
end
Rectangle Nonskewness
By the principle of inclusion and exclusion, rectangle nonskewness is
reduced to 2^{d}  1 instances of rectangle intersection.
Therefore it can be solved with exactly the same table, in time
O(d4^{d}).
4. Lower Bounds
There are exactly 2^{n} generalized intervals and
exactly n(n  1) wrapped intervals in {1, . . . , n}.
Therefore, Generalized Rectangle Intersection can be solved by table
lookup with a table of size 2^{n1}
. . . 2^{nd}, and Wrapped Interval Section can
be solved by table lookup with a table of size
n_{1}(n_{1}  1) . . .
n_{d}(n_{d}  1). We will
show that no smaller tables suffice for either problem, no matter how
much time is allowed.
For simplicity we will consider only the case d=1. In the full
version of this paper we will explain how the general case is a corollary
of this one. Henceforth let n = n_{1}.
Generalized Rectangle Intersection
Suppose that given some table T we can answer questions of the form
``how many elements of D intersect Q?'', where D is a fixed multiset
of generalized intervals and Q is a generalized interval.
Previously, we said that the intersection questions are really
logicalAND questions. Actually, they are ANDs over all dimensions.
But in each single dimension, the question is an OR, i.e., ``does at
least one square of the query interval belong to the dataset
rectangle?''
Let
#(x_{1} or . . . or x_{k}) denote the number of generalized
rectangles r in D such that x_{1} ε r or . . . or x_{k} ε
r. Let #(x_{1} and . . . and x_{k}) denote the number of
generalized rectangles r in D such that x_{1} ε r and . . .
and x_{k} ε r. By assumption, we can compute #(x_{1} or . . .
or x_{k}) from T. By the principal of inclusion and exclusion, we
have
#(x_{1} and x_{2}) = #(x_{1}) + #(x_{2})  #(x_{1} or x_{2}).
Thus we can compute #(x_{1} and x_{2}) from T. By a simple
induction, we can compute #(x_{1} and . . . and x_{k}) from T.
Let #(x_{1} and . . . and x_{k} and ¬ x_{k+1} and . . .
and ¬ x_{m}) denote the number of generalized rectangles r in
D such that x_{1} ε r and . . . and x_{k} ε r and ¬ (x_{k+1} ε
r) and . . . and ¬ (x_{m} ε r). We have
#(x_{1} and . . . and x_{k} and ¬ x_{k+1}) =
#(x_{1} and . . . and x_{k})  #(x_{1} and . . . and x_{k+1}).
Thus we can compute #(x_{1} and . . . and x_{k} and ¬
x_{k+1}) from T. By a simple induction we can compute #(x_{1}
and . . . and x_{k} and ¬ x_{k+1} and . . . and ¬
xn) from T, where {x_{1}, . . . , x_{n}} = {1, . . . , n}. Thus we
can determine from T exactly how many times the generalized
rectangle {x_{1}, . . . , x_{k}} appears in the multiset D. Since we
can recover 2^{n} independent numbers from T, the size of T must
be at least 2^{n}.
Note: a similar argument shows that if we limit the size of generalized
intervals to k then we still can't get by with nonexhaustive tables.
Wrapped Rectangle Intersection
Suppose that given some table T we can answer questions of the form
``how many elements of D intersect Q?'', where D is a fixed multiset
of wrapped intervals and Q is a wrapped interval. If i ≤ j,
let #[i, j] denote the number of elements of D contained in the
interval [i, j]. Then #[i, j] is equal to the number of elements
of D (that intersect [1, n]) minus the number of elements of D
that intersect {j+1, . . . , n, 1, . . . , i  1}, so we can compute
#[i, j] from T.
The number of times that the interval #[i, j] appears in the list D
is given by the formula:
#[i, j]  #[i  1, j]  #[i, j  1] + #[i  1, j  1].
By a similar argument we can determine the number of times each
wrapped interval appears in D. Since we can recover n(n  1) independent
numbers from T, the size of T must be at least n(n  1).
Acknowledgments
This work was supported by NASA grant 52895. The research was
performed while the first author was on sabbatical from Yale
University, visiting the HumanComputer Interaction Laboratory at the
University of Maryland. He is also partially supported by NSF under
grants CCR9700417 and CCR9796317. Both authors are grateful to
Catherine Plaisant and Ben Shneiderman for their part in formulating
this problem and to Dave Mount and Dan Spielman for helpful
discussions.
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