Thrill  0.1
Step 3: ReduceByKey to Calculate New Cluster Centers

In step 2 of this tutorial we constructed a DIA containing each point and its closest center id. The next step of Lloyd's algorithm is to calculate the mean (average) over all points associated with the current cluster center. The mean points will be the new centers in the next iteration.

How to calculate the mean point? In Thrill this can be done using a ReduceByKey() reduction, which is similar to MapReduce's reduce method. The idea is to sum all points associated with a center, and then divide by their number, which results in the average point value.

To accomplish this in the code, we extend the ClosestCenter class with another field: count. The reduction will add the field point and the count field, which together represent an average point.

//! Assignment of a point to a cluster.
struct ClosestCenter {
size_t cluster_id;
Point point;
size_t count;
};
//! make ostream-able for Print()
std::ostream& operator << (std::ostream& os, const ClosestCenter& cc) {
return os << '(' << cc.cluster_id
<< ':' << cc.point << ':' << cc.count << ')';
}

The count field is initialized with 1 in the result of Map() calculation of the closest center.

return ClosestCenter { cluster_id, p, /* count */ 1 };

And then we use ReduceByKey() for the reduction. The ReduceByKey() DIA operation requires two lambda methods: a key extractor and a reduction function. The key extractor is simply the cluster_id field, since this is by which we want the set of point associations to be grouped.

// calculate new centers as the mean of all points associated with its id
auto reduced_centers =
closest
.ReduceByKey(
// key extractor: the cluster id
[](const ClosestCenter& cc) { return cc.cluster_id; },
// reduction: add points and the counter
[](const ClosestCenter& a, const ClosestCenter& b) {
return ClosestCenter {
a.cluster_id, a.point + b.point, a.count + b.count
};
});
reduced_centers.Print("reduced_centers");

The reduction function "adds" two ClosestCenter structs by adding the point and count fields, but keeping the cluster_id constant.

In step 3's code we added a .Print() for debugging the reduction.

auto new_centers =
reduced_centers
.Map([](const ClosestCenter& cc) {
return cc.point / cc.count;
});
new_centers.Print("new_centers");

After the reduction, we must divide by the total points aggregated to deliver the new average centers. This is a simple application of Map() which takes a ClosestCenter and returns a Point.

The resulting new_centers is a DIA<Point> containing the next iteration's cluster centers.

We implicitly used some vector operators on Point inside the reduction: plus and scalar division. Again, with high-flying object-oriented spirits, we extended the Point class with the appropriate operators, which make the functions above very readable.

//! A 2-dimensional point with double precision
struct Point {
//! point coordinates
double x, y;
double DistanceSquare(const Point& b) const {
return (x - b.x) * (x - b.x) + (y - b.y) * (y - b.y);
}
Point operator + (const Point& b) const {
return Point { x + b.x, y + b.y };
}
Point operator / (double s) const {
return Point { x / s, y / s };
}
};

See the complete example code examples/tutorial/k-means_step3.cpp

The output of our program so far is something like the following:

[... as before ...]
[host 0 worker 0 000011] PushData() stage Cache.2 with targets [ReduceByKey.8]
[host 0 worker 0 000012] Execute()  stage ReduceByKey.8
[host 0 worker 0 000013] PushData() stage ReduceByKey.8 with targets [Print.9]
[host 0 worker 0 000014] Execute()  stage Print.9
reduced_centers --- Begin DIA.Print() --- size=10
reduced_centers: (0:(63.1991,406.621):2)
reduced_centers: (1:(4152.89,999.313):5)
reduced_centers: (7:(6394.55,2896.39):7)
reduced_centers: (5:(17904.5,18761.1):24)
reduced_centers: (4:(5667.67,3197.76):9)
reduced_centers: (9:(1842.71,6814.27):8)
reduced_centers: (2:(3620.52,9055.77):16)
reduced_centers: (8:(1473.69,2804.14):11)
reduced_centers: (3:(7621.35,1216.32):13)
reduced_centers: (6:(903.419,125.585):5)
reduced_centers --- End DIA.Print() --- size=10
[host 0 worker 0 000015] PushData() stage ReduceByKey.8 with targets [Print.11]
[host 0 worker 0 000016] Execute()  stage Print.11
new_centers --- Begin DIA.Print() --- size=10
new_centers: (31.5996,203.311)
new_centers: (830.578,199.863)
new_centers: (913.507,413.77)
new_centers: (746.021,781.712)
new_centers: (629.741,355.307)
new_centers: (230.339,851.784)
new_centers: (226.283,565.986)
new_centers: (133.972,254.921)
new_centers: (586.258,93.5627)
new_centers: (180.684,25.1171)
new_centers --- End DIA.Print() --- size=10
[...]