path: root/swarm/network/kademlia.go

// Copyright 2017 The go-ethereum Authors
// This file is part of the go-ethereum library.
//
// The go-ethereum library is free software: you can redistribute it and/or modify
// it under the terms of the GNU Lesser General Public License as published by
// the Free Software Foundation, either version 3 of the License, or
// (at your option) any later version.
//
// The go-ethereum library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
// GNU Lesser General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public License
// along with the go-ethereum library. If not, see <http://www.gnu.org/licenses/>.

package network

import (
    "bytes"
    "fmt"
    "math/rand"
    "strings"
    "sync"
    "time"

    "github.com/ethereum/go-ethereum/common"
    "github.com/ethereum/go-ethereum/swarm/log"
    "github.com/ethereum/go-ethereum/swarm/pot"
)

/*

Taking the proximity order relative to a fix point x classifies the points in
the space (n byte long byte sequences) into bins. Items in each are at
most half as distant from x as items in the previous bin. Given a sample of
uniformly distributed items (a hash function over arbitrary sequence) the
proximity scale maps onto series of subsets with cardinalities on a negative
exponential scale.

It also has the property that any two item belonging to the same bin are at
most half as distant from each other as they are from x.

If we think of random sample of items in the bins as connections in a network of
interconnected nodes then relative proximity can serve as the basis for local
decisions for graph traversal where the task is to find a route between two
points. Since in every hop, the finite distance halves, there is
a guaranteed constant maximum limit on the number of hops needed to reach one
node from the other.
*/

var pof = pot.DefaultPof(256)

// KadParams holds the config params for Kademlia
type KadParams struct {
    // adjustable parameters
    MaxProxDisplay int   // number of rows the table shows
    MinProxBinSize int   // nearest neighbour core minimum cardinality
    MinBinSize     int   // minimum number of peers in a row
    MaxBinSize     int   // maximum number of peers in a row before pruning
    RetryInterval  int64 // initial interval before a peer is first redialed
    RetryExponent  int   // exponent to multiply retry intervals with
    MaxRetries     int   // maximum number of redial attempts
    // function to sanction or prevent suggesting a peer
    Reachable func(OverlayAddr) bool
}

// NewKadParams returns a params struct with default values
func NewKadParams() *KadParams {
    return &KadParams{
        MaxProxDisplay: 16,
        MinProxBinSize: 2,
        MinBinSize:     2,
        MaxBinSize:     4,
        RetryInterval:  4200000000, // 4.2 sec
        MaxRetries:     42,
        RetryExponent:  2,
    }
}

// Kademlia is a table of live peers and a db of known peers (node records)
type Kademlia struct {
    lock       sync.RWMutex
    *KadParams          // Kademlia configuration parameters
    base       []byte   // immutable baseaddress of the table
    addrs      *pot.Pot // pots container for known peer addresses
    conns      *pot.Pot // pots container for live peer connections
    depth      uint8    // stores the last current depth of saturation
    nDepth     int      // stores the last neighbourhood depth
    nDepthC    chan int // returned by DepthC function to signal neighbourhood depth change
    addrCountC chan int // returned by AddrCountC function to signal peer count change
}

// NewKademlia creates a Kademlia table for base address addr
// with parameters as in params
// if params is nil, it uses default values
func NewKademlia(addr []byte, params *KadParams) *Kademlia {
    if params == nil {
        params = NewKadParams()
    }
    return &Kademlia{
        base:      addr,
        KadParams: params,
        addrs:     pot.NewPot(nil, 0),
        conns:     pot.NewPot(nil, 0),
    }
}

// OverlayPeer interface captures the common aspect of view of a peer from the Overlay
// topology driver
type OverlayPeer interface {
    Address() []byte
}

// OverlayConn represents a connected peer
type OverlayConn interface {
    OverlayPeer
    Drop(error)       // call to indicate a peer should be expunged
    Off() OverlayAddr // call to return a persitent OverlayAddr
}

// OverlayAddr represents a kademlia peer record
type OverlayAddr interface {
    OverlayPeer
    Update(OverlayAddr) OverlayAddr // returns the updated version of the original
}

// entry represents a Kademlia table entry (an extension of OverlayPeer)
type entry struct {
    OverlayPeer
    seenAt  time.Time
    retries int
}

// newEntry creates a kademlia peer from an OverlayPeer interface
func newEntry(p OverlayPeer) *entry {
    return &entry{
        OverlayPeer: p,
        seenAt:      time.Now(),
    }
}

// Bin is the binary (bitvector) serialisation of the entry address
func (e *entry) Bin() string {
    return pot.ToBin(e.addr().Address())
}

// Label is a short tag for the entry for debug
func Label(e *entry) string {
    return fmt.Sprintf("%s (%d)", e.Hex()[:4], e.retries)
}

// Hex is the hexadecimal serialisation of the entry address
func (e *entry) Hex() string {
    return fmt.Sprintf("%x", e.addr().Address())
}

// String is the short tag for the entry
func (e *entry) String() string {
    return fmt.Sprintf("%s (%d)", e.Hex()[:8], e.retries)
}

// addr returns the kad peer record (OverlayAddr) corresponding to the entry
func (e *entry) addr() OverlayAddr {
    a, _ := e.OverlayPeer.(OverlayAddr)
    return a
}

// conn returns the connected peer (OverlayPeer) corresponding to the entry
func (e *entry) conn() OverlayConn {
    c, _ := e.OverlayPeer.(OverlayConn)
    return c
}

// Register enters each OverlayAddr as kademlia peer record into the
// database of known peer addresses
func (k *Kademlia) Register(peers []OverlayAddr) error {
    k.lock.Lock()
    defer k.lock.Unlock()
    var known, size int
    for _, p := range peers {
        // error if self received, peer should know better
        // and should be punished for this
        if bytes.Equal(p.Address(), k.base) {
            return fmt.Errorf("add peers: %x is self", k.base)
        }
        var found bool
        k.addrs, _, found, _ = pot.Swap(k.addrs, p, pof, func(v pot.Val) pot.Val {
            // if not found
            if v == nil {
                // insert new offline peer into conns
                return newEntry(p)
            }
            // found among known peers, do nothing
            return v
        })
        if found {
            known++
        }
        size++
    }
    // send new address count value only if there are new addresses
    if k.addrCountC != nil && size-known > 0 {
        k.addrCountC <- k.addrs.Size()
    }
    // log.Trace(fmt.Sprintf("%x registered %v peers, %v known, total: %v", k.BaseAddr()[:4], size, known, k.addrs.Size()))

    k.sendNeighbourhoodDepthChange()
    return nil
}

// SuggestPeer returns a known peer for the lowest proximity bin for the
// lowest bincount below depth
// naturally if there is an empty row it returns a peer for that
func (k *Kademlia) SuggestPeer() (a OverlayAddr, o int, want bool) {
    k.lock.Lock()
    defer k.lock.Unlock()
    minsize := k.MinBinSize
    depth := k.neighbourhoodDepth()
    // if there is a callable neighbour within the current proxBin, connect
    // this makes sure nearest neighbour set is fully connected
    var ppo int
    k.addrs.EachNeighbour(k.base, pof, func(val pot.Val, po int) bool {
        if po < depth {
            return false
        }
        a = k.callable(val)
        ppo = po
        return a == nil
    })
    if a != nil {
        log.Trace(fmt.Sprintf("%08x candidate nearest neighbour found: %v (%v)", k.BaseAddr()[:4], a, ppo))
        return a, 0, false
    }
    // log.Trace(fmt.Sprintf("%08x no candidate nearest neighbours to connect to (Depth: %v, minProxSize: %v) %#v", k.BaseAddr()[:4], depth, k.MinProxBinSize, a))

    var bpo []int
    prev := -1
    k.conns.EachBin(k.base, pof, 0, func(po, size int, f func(func(val pot.Val, i int) bool) bool) bool {
        prev++
        for ; prev < po; prev++ {
            bpo = append(bpo, prev)
            minsize = 0
        }
        if size < minsize {
            bpo = append(bpo, po)
            minsize = size
        }
        return size > 0 && po < depth
    })
    // all buckets are full, ie., minsize == k.MinBinSize
    if len(bpo) == 0 {
        // log.Debug(fmt.Sprintf("%08x: all bins saturated", k.BaseAddr()[:4]))
        return nil, 0, false
    }
    // as long as we got candidate peers to connect to
    // dont ask for new peers (want = false)
    // try to select a candidate peer
    // find the first callable peer
    nxt := bpo[0]
    k.addrs.EachBin(k.base, pof, nxt, func(po, _ int, f func(func(pot.Val, int) bool) bool) bool {
        // for each bin (up until depth) we find callable candidate peers
        if po >= depth {
            return false
        }
        return f(func(val pot.Val, _ int) bool {
            a = k.callable(val)
            return a == nil
        })
    })
    // found a candidate
    if a != nil {
        return a, 0, false
    }
    // no candidate peer found, request for the short bin
    var changed bool
    if uint8(nxt) < k.depth {
        k.depth = uint8(nxt)
        changed = true
    }
    return a, nxt, changed
}

// On inserts the peer as a kademlia peer into the live peers
func (k *Kademlia) On(p OverlayConn) (uint8, bool) {
    k.lock.Lock()
    defer k.lock.Unlock()
    e := newEntry(p)
    var ins bool
    k.conns, _, _, _ = pot.Swap(k.conns, p, pof, func(v pot.Val) pot.Val {
        // if not found live
        if v == nil {
            ins = true
            // insert new online peer into conns
            return e
        }
        // found among live peers, do nothing
        return v
    })
    if ins {
        // insert new online peer into addrs
        k.addrs, _, _, _ = pot.Swap(k.addrs, p, pof, func(v pot.Val) pot.Val {
            return e
        })
        // send new address count value only if the peer is inserted
        if k.addrCountC != nil {
            k.addrCountC <- k.addrs.Size()
        }
    }
    log.Trace(k.string())
    // calculate if depth of saturation changed
    depth := uint8(k.saturation(k.MinBinSize))
    var changed bool
    if depth != k.depth {
        changed = true
        k.depth = depth
    }
    k.sendNeighbourhoodDepthChange()
    return k.depth, changed
}

// NeighbourhoodDepthC returns the channel that sends a new kademlia
// neighbourhood depth on each change.
// Not receiving from the returned channel will block On function
// when the neighbourhood depth is changed.
func (k *Kademlia) NeighbourhoodDepthC() <-chan int {
    if k.nDepthC == nil {
        k.nDepthC = make(chan int)
    }
    return k.nDepthC
}

// sendNeighbourhoodDepthChange sends new neighbourhood depth to k.nDepth channel
// if it is initialized.
func (k *Kademlia) sendNeighbourhoodDepthChange() {
    // nDepthC is initialized when NeighbourhoodDepthC is called and returned by it.
    // It provides signaling of neighbourhood depth change.
    // This part of the code is sending new neighbourhood depth to nDepthC if that condition is met.
    if k.nDepthC != nil {
        nDepth := k.neighbourhoodDepth()
        if nDepth != k.nDepth {
            k.nDepth = nDepth
            k.nDepthC <- nDepth
        }
    }
}

// AddrCountC returns the channel that sends a new
// address count value on each change.
// Not receiving from the returned channel will block Register function
// when address count value changes.
func (k *Kademlia) AddrCountC() <-chan int {
    if k.addrCountC == nil {
        k.addrCountC = make(chan int)
    }
    return k.addrCountC
}

// Off removes a peer from among live peers
func (k *Kademlia) Off(p OverlayConn) {
    k.lock.Lock()
    defer k.lock.Unlock()
    var del bool
    k.addrs, _, _, _ = pot.Swap(k.addrs, p, pof, func(v pot.Val) pot.Val {
        // v cannot be nil, must check otherwise we overwrite entry
        if v == nil {
            panic(fmt.Sprintf("connected peer not found %v", p))
        }
        del = true
        return newEntry(p.Off())
    })

    if del {
        k.conns, _, _, _ = pot.Swap(k.conns, p, pof, func(_ pot.Val) pot.Val {
            // v cannot be nil, but no need to check
            return nil
        })
        // send new address count value only if the peer is deleted
        if k.addrCountC != nil {
            k.addrCountC <- k.addrs.Size()
        }
        k.sendNeighbourhoodDepthChange()
    }
}

func (k *Kademlia) EachBin(base []byte, pof pot.Pof, o int, eachBinFunc func(conn OverlayConn, po int) bool) {
    k.lock.RLock()
    defer k.lock.RUnlock()

    var startPo int
    var endPo int
    kadDepth := k.neighbourhoodDepth()

    k.conns.EachBin(base, pof, o, func(po, size int, f func(func(val pot.Val, i int) bool) bool) bool {
        if startPo > 0 && endPo != k.MaxProxDisplay {
            startPo = endPo + 1
        }
        if po < kadDepth {
            endPo = po
        } else {
            endPo = k.MaxProxDisplay
        }

        for bin := startPo; bin <= endPo; bin++ {
            f(func(val pot.Val, _ int) bool {
                return eachBinFunc(val.(*entry).conn(), bin)
            })
        }
        return true
    })
}

// EachConn is an iterator with args (base, po, f) applies f to each live peer
// that has proximity order po or less as measured from the base
// if base is nil, kademlia base address is used
func (k *Kademlia) EachConn(base []byte, o int, f func(OverlayConn, int, bool) bool) {
    k.lock.RLock()
    defer k.lock.RUnlock()
    k.eachConn(base, o, f)
}

func (k *Kademlia) eachConn(base []byte, o int, f func(OverlayConn, int, bool) bool) {
    if len(base) == 0 {
        base = k.base
    }
    depth := k.neighbourhoodDepth()
    k.conns.EachNeighbour(base, pof, func(val pot.Val, po int) bool {
        if po > o {
            return true
        }
        return f(val.(*entry).conn(), po, po >= depth)
    })
}

// EachAddr called with (base, po, f) is an iterator applying f to each known peer
// that has proximity order po or less as measured from the base
// if base is nil, kademlia base address is used
func (k *Kademlia) EachAddr(base []byte, o int, f func(OverlayAddr, int, bool) bool) {
    k.lock.RLock()
    defer k.lock.RUnlock()
    k.eachAddr(base, o, f)
}

func (k *Kademlia) eachAddr(base []byte, o int, f func(OverlayAddr, int, bool) bool) {
    if len(base) == 0 {
        base = k.base
    }
    depth := k.neighbourhoodDepth()
    k.addrs.EachNeighbour(base, pof, func(val pot.Val, po int) bool {
        if po > o {
            return true
        }
        return f(val.(*entry).addr(), po, po >= depth)
    })
}

// neighbourhoodDepth returns the proximity order that defines the distance of
// the nearest neighbour set with cardinality >= MinProxBinSize
// if there is altogether less than MinProxBinSize peers it returns 0
// caller must hold the lock
func (k *Kademlia) neighbourhoodDepth() (depth int) {
    if k.conns.Size() < k.MinProxBinSize {
        return 0
    }
    var size int
    f := func(v pot.Val, i int) bool {
        size++
        depth = i
        return size < k.MinProxBinSize
    }
    k.conns.EachNeighbour(k.base, pof, f)
    return depth
}

// callable when called with val,
func (k *Kademlia) callable(val pot.Val) OverlayAddr {
    e := val.(*entry)
    // not callable if peer is live or exceeded maxRetries
    if e.conn() != nil || e.retries > k.MaxRetries {
        return nil
    }
    // calculate the allowed number of retries based on time lapsed since last seen
    timeAgo := int64(time.Since(e.seenAt))
    div := int64(k.RetryExponent)
    div += (150000 - rand.Int63n(300000)) * div / 1000000
    var retries int
    for delta := timeAgo; delta > k.RetryInterval; delta /= div {
        retries++
    }
    // this is never called concurrently, so safe to increment
    // peer can be retried again
    if retries < e.retries {
        log.Trace(fmt.Sprintf("%08x: %v long time since last try (at %v) needed before retry %v, wait only warrants %v", k.BaseAddr()[:4], e, timeAgo, e.retries, retries))
        return nil
    }
    // function to sanction or prevent suggesting a peer
    if k.Reachable != nil && !k.Reachable(e.addr()) {
        log.Trace(fmt.Sprintf("%08x: peer %v is temporarily not callable", k.BaseAddr()[:4], e))
        return nil
    }
    e.retries++
    log.Trace(fmt.Sprintf("%08x: peer %v is callable", k.BaseAddr()[:4], e))

    return e.addr()
}

// BaseAddr return the kademlia base address
func (k *Kademlia) BaseAddr() []byte {
    return k.base
}

// String returns kademlia table + kaddb table displayed with ascii
func (k *Kademlia) String() string {
    k.lock.RLock()
    defer k.lock.RUnlock()
    return k.string()
}

// String returns kademlia table + kaddb table displayed with ascii
func (k *Kademlia) string() string {
    wsrow := "                          "
    var rows []string

    rows = append(rows, "=========================================================================")
    rows = append(rows, fmt.Sprintf("%v KΛÐΞMLIΛ hive: queen's address: %x", time.Now().UTC().Format(time.UnixDate), k.BaseAddr()[:3]))
    rows = append(rows, fmt.Sprintf("population: %d (%d), MinProxBinSize: %d, MinBinSize: %d, MaxBinSize: %d", k.conns.Size(), k.addrs.Size(), k.MinProxBinSize, k.MinBinSize, k.MaxBinSize))

    liverows := make([]string, k.MaxProxDisplay)
    peersrows := make([]string, k.MaxProxDisplay)

    depth := k.neighbourhoodDepth()
    rest := k.conns.Size()
    k.conns.EachBin(k.base, pof, 0, func(po, size int, f func(func(val pot.Val, i int) bool) bool) bool {
        var rowlen int
        if po >= k.MaxProxDisplay {
            po = k.MaxProxDisplay - 1
        }
        row := []string{fmt.Sprintf("%2d", size)}
        rest -= size
        f(func(val pot.Val, vpo int) bool {
            e := val.(*entry)
            row = append(row, fmt.Sprintf("%x", e.Address()[:2]))
            rowlen++
            return rowlen < 4
        })
        r := strings.Join(row, " ")
        r = r + wsrow
        liverows[po] = r[:31]
        return true
    })

    k.addrs.EachBin(k.base, pof, 0, func(po, size int, f func(func(val pot.Val, i int) bool) bool) bool {
        var rowlen int
        if po >= k.MaxProxDisplay {
            po = k.MaxProxDisplay - 1
        }
        if size < 0 {
            panic("wtf")
        }
        row := []string{fmt.Sprintf("%2d", size)}
        // we are displaying live peers too
        f(func(val pot.Val, vpo int) bool {
            e := val.(*entry)
            row = append(row, Label(e))
            rowlen++
            return rowlen < 4
        })
        peersrows[po] = strings.Join(row, " ")
        return true
    })

    for i := 0; i < k.MaxProxDisplay; i++ {
        if i == depth {
            rows = append(rows, fmt.Sprintf("============ DEPTH: %d ==========================================", i))
        }
        left := liverows[i]
        right := peersrows[i]
        if len(left) == 0 {
            left = " 0                             "
        }
        if len(right) == 0 {
            right = " 0"
        }
        rows = append(rows, fmt.Sprintf("%03d %v | %v", i, left, right))
    }
    rows = append(rows, "=========================================================================")
    return "\n" + strings.Join(rows, "\n")
}

// PeerPot keeps info about expected nearest neighbours and empty bins
// used for testing only
type PeerPot struct {
    NNSet     [][]byte
    EmptyBins []int
}

// NewPeerPotMap creates a map of pot record of OverlayAddr with keys
// as hexadecimal representations of the address.
func NewPeerPotMap(kadMinProxSize int, addrs [][]byte) map[string]*PeerPot {
    // create a table of all nodes for health check
    np := pot.NewPot(nil, 0)
    for _, addr := range addrs {
        np, _, _ = pot.Add(np, addr, pof)
    }
    ppmap := make(map[string]*PeerPot)

    for i, a := range addrs {
        pl := 256
        prev := 256
        var emptyBins []int
        var nns [][]byte
        np.EachNeighbour(addrs[i], pof, func(val pot.Val, po int) bool {
            a := val.([]byte)
            if po == 256 {
                return true
            }
            if pl == 256 || pl == po {
                nns = append(nns, a)
            }
            if pl == 256 && len(nns) >= kadMinProxSize {
                pl = po
                prev = po
            }
            if prev < pl {
                for j := prev; j > po; j-- {
                    emptyBins = append(emptyBins, j)
                }
            }
            prev = po - 1
            return true
        })
        for j := prev; j >= 0; j-- {
            emptyBins = append(emptyBins, j)
        }
        log.Trace(fmt.Sprintf("%x NNS: %s", addrs[i][:4], LogAddrs(nns)))
        ppmap[common.Bytes2Hex(a)] = &PeerPot{nns, emptyBins}
    }
    return ppmap
}

// saturation returns the lowest proximity order that the bin for that order
// has less than n peers
func (k *Kademlia) saturation(n int) int {
    prev := -1
    k.addrs.EachBin(k.base, pof, 0, func(po, size int, f func(func(val pot.Val, i int) bool) bool) bool {
        prev++
        return prev == po && size >= n
    })
    depth := k.neighbourhoodDepth()
    if depth < prev {
        return depth
    }
    return prev
}

// full returns true if all required bins have connected peers.
// It is used in Healthy function.
func (k *Kademlia) full(emptyBins []int) (full bool) {
    prev := 0
    e := len(emptyBins)
    ok := true
    depth := k.neighbourhoodDepth()
    k.conns.EachBin(k.base, pof, 0, func(po, _ int, _ func(func(val pot.Val, i int) bool) bool) bool {
        if prev == depth+1 {
            return true
        }
        for i := prev; i < po; i++ {
            e--
            if e < 0 {
                ok = false
                return false
            }
            if emptyBins[e] != i {
                log.Trace(fmt.Sprintf("%08x po: %d, i: %d, e: %d, emptybins: %v", k.BaseAddr()[:4], po, i, e, logEmptyBins(emptyBins)))
                if emptyBins[e] < i {
                    panic("incorrect peerpot")
                }
                ok = false
                return false
            }
        }
        prev = po + 1
        return true
    })
    if !ok {
        return false
    }
    return e == 0
}

func (k *Kademlia) knowNearestNeighbours(peers [][]byte) bool {
    pm := make(map[string]bool)

    k.eachAddr(nil, 255, func(p OverlayAddr, po int, nn bool) bool {
        if !nn {
            return false
        }
        pk := fmt.Sprintf("%x", p.Address())
        pm[pk] = true
        return true
    })
    for _, p := range peers {
        pk := fmt.Sprintf("%x", p)
        if !pm[pk] {
            log.Trace(fmt.Sprintf("%08x: known nearest neighbour %s not found", k.BaseAddr()[:4], pk[:8]))
            return false
        }
    }
    return true
}

func (k *Kademlia) gotNearestNeighbours(peers [][]byte) (got bool, n int, missing [][]byte) {
    pm := make(map[string]bool)

    k.eachConn(nil, 255, func(p OverlayConn, po int, nn bool) bool {
        if !nn {
            return false
        }
        pk := fmt.Sprintf("%x", p.Address())
        pm[pk] = true
        return true
    })
    var gots int
    var culprits [][]byte
    for _, p := range peers {
        pk := fmt.Sprintf("%x", p)
        if pm[pk] {
            gots++
        } else {
            log.Trace(fmt.Sprintf("%08x: ExpNN: %s not found", k.BaseAddr()[:4], pk[:8]))
            culprits = append(culprits, p)
        }
    }
    return gots == len(peers), gots, culprits
}

// Health state of the Kademlia
type Health struct {
    KnowNN     bool     // whether node knows all its nearest neighbours
    GotNN      bool     // whether node is connected to all its nearest neighbours
    CountNN    int      // amount of nearest neighbors connected to
    CulpritsNN [][]byte // which known NNs are missing
    Full       bool     // whether node has a peer in each kademlia bin (where there is such a peer)
    Hive       string
}

// Healthy reports the health state of the kademlia connectivity
// returns a Health struct
func (k *Kademlia) Healthy(pp *PeerPot) *Health {
    k.lock.RLock()
    defer k.lock.RUnlock()
    gotnn, countnn, culpritsnn := k.gotNearestNeighbours(pp.NNSet)
    knownn := k.knowNearestNeighbours(pp.NNSet)
    full := k.full(pp.EmptyBins)
    log.Trace(fmt.Sprintf("%08x: healthy: knowNNs: %v, gotNNs: %v, full: %v\n", k.BaseAddr()[:4], knownn, gotnn, full))
    return &Health{knownn, gotnn, countnn, culpritsnn, full, k.string()}
}

func logEmptyBins(ebs []int) string {
    var ebss []string
    for _, eb := range ebs {
        ebss = append(ebss, fmt.Sprintf("%d", eb))
    }
    return strings.Join(ebss, ", ")
}